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Heritable Edition

Topology of Infinite-Dimensional

Manifolds

Katsuro Sakai

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June 6, 2016

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Topology of Inﬁnite-Dimensional Manifolds

⃝2015 by Katsuro Sakaic

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Katsuro Sakai

Topology of Infinite-Dimensional

Manifolds

June 6, 2016

Springer

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Springer Monographs of Mathematics

Author: Katsuro Sakai

Title: Topology of Inﬁnite-Dimensional Manifolds MSC2010 : 57-01, 57-02, 57N20, 54F65, 54C55

Keywords and phrase : Manifolds modeled on a space,

Hilbert space, Hilbert cube, Universal spaces for absolute Borel classes, The direct limit of Euclidean spaces, The direct limit of Hilbert cubes, The bounded weak-star topology

⃝2015 by Katsuro Sakaic

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Because we are made in God’s image, we can thinking and imagining various things indeﬁnitely. We can also consider abstract concepts. Mathe- matical ability is a gift from God. Indeed, what we do have that we did not received? There is no reason why we boast (1 Corinthians 4:7). Instead, we always thank and praise Jehovah God, our Creator. All of us are the work of Jehovah’s hands. I agree with the following phrase in the Scriptures:

The conclusion of the matter, everything having been heard, is: Fear the true God and keep his commandments, for this is the whole obligation of man. For the true God will judge every deed, including every hidden thing, as to whether it is good or bad.

— Ecclesiastes 12:13, 14 So, we should always keep this phrase in mind:

The fear of Jehovah is the beginning of wisdom, and knowledge of the Most Holly One is understanding. — in Proverbs 9:10

Note: Scripture quotations are from the modern-language New World Translation of the Holy Scriptures

— 2013 Version

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Preface

This is a textbook for graduate students acquiring fundamental knowledge of inﬁnite-dimensional manifolds. Up to now, the following six books are avail- able for the same purpose:

(1) C. Bessaga and A. Pe lczy´nski, Selected Topics in Infinite-Dimensional Topology, MM 58 (Polish Sci. Publ., Warsaw, 1975)

(2) T.A. Chapman, Lectures on Hilbert Cube Manifolds, CBMS Regional Conf. Ser. in Math. 28 (Amer. Math. Soc., Providence, 1975)

(3) J. van Mill, Infinite-Dimensional Topology, Prerequisites and Introduction, North-Holland Math. Library 43, (Elsevier Sci. Publ. B.V., Amsterdam, 1989)

(4) A. Chigogidze, Inverse Spectra, North-Holland Math. Library 53, (Else- vier Sci. Publ. B.V., Amsterdam, 1996)

(5) T. Banakh, T. Radul and M. Zarichnyi, Absorbing Sets in Infinite- Dimensional Manifolds, Math. Studies, Monog. Ser. 1 (VNTL Publ., Lviv, 1996)

(6) J. van Mill, The Infinite-Dimensional Topology of Function Spaces, North- Holland Math. Library 64, (Elsevier Sci. Publ. B.V., Amsterdam, 2002) Selecting contents of these books and adding new results which are not contained in them, the present book is written without assuming separability. We also treat manifolds modeled on the direct limits of Euclidean spaces and Hilbert cubes, which are teated in no other texts.

Readers are required to ﬁnish the graduate course on General Topology. For example, concerning General Topology, it is enough to ﬁnish Part I of the following textbook that is popular:

• J.R. Munkres, Topology, 2nd ed. (Prentice Hall, Inc., Upper Saddle River, 2000)

However, some additional results on General Topology are necessary except the contents of the above book. For instance, the concept of the weak topol-

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ogy, the direct (inductive) limit of spaces, etc. Such additional results can be founded in the 2nd chapter of the following author’s book:

• K. Sakai, Geometric Aspects of General Topology, Springer Monog. in Math. (Springer, Tokyo, 2013)

Hereafter, this is cited as [GAGT]. Besides, ANR Theory (Theory of Retracts) is indispensable. In addition, some knowledge of topological linear spaces, simplicial complexes and dimensions are also necessary. These required knowledge can be also obtained in the book [GAGT]. Required background knowledge are listed in Chap. 1.

To learn ANR theory, two classical textbooks by K. Borsuk and by S.- T. Hu, with the same title are still excellent:

• K. Borsuk, Theory of Retracts, MM 44 (Polish Sci. Publ., Warsaw, 1967)

• S.-T. Hu, Theory of Retracts (Wayne State University Press, Detroit, 1965) After publication of these books, the theory has been greatly developed. The ﬁfth chapter of van Mill’s ﬁrst book (3) is also excellent to study elements of ANRs but its content is the almost same as these books. The 6th chapter of [GAGT] contains up-dated results. If the reader has some of them, then he refers to this book only for unfamiliar results.

Chapter 2 is devoted to fundamental results on infinite-dimensional manifolds modeled on an infinite-dimensional locally convex metrizable topological linear space, the Stability Theorem, the Unknotting Theorem, the Open Embedding Theorem, the Classification Theorem, etc., which are discussed in Bessaga-Pe lczynski’s book (1). Here is also proved fundamental results on Hilbert cube manifolds, which are discussed in van Mill’s book (3) and Chapman’s lecture notes (2). Furthermore, we prove the Toruńczyk Factor Theorem, which has not been written in any other book.

In Chap. 3, we give proofs of Toru´nczyk’s characterizations of Hilbert manifolds and Hilbert cube manifolds. For the characterization of compact Hilbert cube manifolds, a readable proof is provided in van Mill’s book (3) but the non-compact version is not easily derived from the compact case. The non-compact case is discussed here. We also discuss non-separable Hilbert manifolds.

We discuss in Chap. 4 absorbing sets in Hilbert manifolds, which are treated in the book of Banakh, Radul, and Zarichinyi (5), but we do not assume the separability. Here, we give the characterization of manifolds modeled on the universal spaces of absolute Borel spaces, that is a generalization of the work of Bestvina and Mogilski to non-separable spaces.

In Chap. 5, we characterize manifolds modeled on the direct limits of Eu- clidean spaces and Hilbert cubes, and prove their Classiﬁcation Theorems. We also discussing simplicial complexes triangulating manifolds modeled on the direct limits of Euclidean spaces. Such a simplicial complex is an inﬁnite- dimensional generalization of combinatorial manifolds, which is called an

ii

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infinite-dimensional combinatorial manifold. We prove so-called Hauptvermu- tung for infinite-dimensional combinatorial manifolds. The inductive limit of Fréchet spaces in the category of topological linear spaces is called LF spaces and manifolds modeled on LF spaces are called LF manifolds. The direct limits of Euclidean spaces is an LF space. Topology of LF manifolds is an interesting subject, but we only introduce some resent results.

The author would like to express his sincere appreciation to his teacher, Professor Yukihiro Kodama, who introduced him to Inﬁnite-Dimensional Topology and had been encouraging him to study in this ﬁeld. He truly thanks his graduate students, Yutaka Iwamoto, Shigenori Uehara, Masayuki Kuri- hara, Masato Yaguchi, Kotaro Mine, Atsushi Yamashita, Atsushi Kogasaka, and Katsuhisa Koshino, for their careful reading his lecture notes and helpful comments. He also fells grateful to Tatsukhiko Yagasaki for encouraging him to publsih this book.

Katsuro Sakai 2016, Tsukuba

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1 Preliminaries and Background Results

In this chapter, we first introduce the terminology and notation, and then list up background results (some specific background results will be listed up in the first section of each chapter). Several results might be learned in graduate course, but others are advanced and special. Almost all results are listed without proofs. Since they can be found in the following author’s book (cited as [GAGT]), their proofs can be confirmed by that book:

K. Sakai, Geometric Aspects of General Topology, Springer Monog. in Math. (Springer, Tokyo, 2013)

For easy of reference, the theorem numbers in [GAGT] are indicated by the bracketed number in the side margin.

1.1 Terminology and Notation

With respect to terminology and notation, we follow the book [GAGT]. For the standard sets, we use the following notation:

• N — the set of natural numbers (i.e., positive integers);

• ω =N ∪ {0} — the set of non-negative integers;

• Z — the set of integers;

• Q — the set of rationals;

• R = (−∞, ∞) — the real line with the usual topology;

• C — the complex plane;

• _R+= [0, ∞) — the half (real) line;

• I = [0, 1] — the unit closed interval.

A (topological) space is assumed to be Hausdorff and a map is a continuous function. A singleton is a space consisting of one point, which is also said to be degenerate. A space is said to be non-degenerate if it is not a singleton. A compact metrizable space is called a compactum and a

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2 1 Preliminaries and Background Results

connected compactum is called a continuum.¹ For a space X and A ⊂ X, we use the following notation:

• clXA (or cl A) — the closure of A in X;

• intXA (or int A) — the interior of A in X;

• bdXA (or bd A) — the boundary of A in X;

• idX (or id) — the identity map of X. For a metrizable space X,

• Metr(X) — the set of all admissible metrics of X.

For a set Γ , the cardinality of Γ is denoted by card Γ . The weight w(X), the density dens X, and the cellularity c(X) of a space X are deﬁned as follows:

• w(X) = min{card B | B is an open basis for X};

• dens X = min{card D | D is a dense set in X};

• c(X) = sup{card G | G is a pair-wise disjoint open collection}.

As is easily observed, c(X) 6 dens X 6 w(X) in general. In case X is metrizable, all these cardinalities coincide.²

For spaces X and Y ,

• X ≈ Y means that X and Y are homeomorphic. Given subspaces X1, . . . , Xn ⊂ X and Y1, . . . , Yn ⊂ Y ,

• (X, X1, . . . , Xn) ≈ (Y, Y1, . . . , Yn) means that there exists a homeomorphism h : X → Y such that h(X1) = Y1, . . . , h(Xn) = Yn;

• (X, x0) ≈ (Y, y0) means (X, {x0}) ≈ (Y, {y0}).

We call (X, x0) a pointed space and x0 its base point.

For the product space ^∏_γ∈Γ Xγ, the γ-coordinate of each point x ∈

∏

γ∈Γ^X^γ is denoted by x(γ), that is, x = (x(γ))γ∈Γ. For each γ ∈ Γ , the projection pr_γ : ^∏_γ∈Γ Xγ → Xγ is defined by pr_γ(x) = x(γ). For Λ ⊂ Γ , the projection pr_Λ : ^∏_γ∈ΓXγ → ^∏_λ∈ΛXλ is defined by pr_Λ(x) = x|Λ (= (x(λ))λ∈Λ). In the case Xγ = X for every γ ∈ Γ , we write^∏_γ∈ΓXγ = X^Γ. In particular, X^Nis the product space of countable infinite copies of X. When Γ = {1, . . . , n}, X^Γ = Xⁿ is the product space of n copies of X. For the product space X × Y , pr_X : X × Y → X and pr_Y : X × Y → Y denote the projections.

Now, let X = (X, d) be a metric space, x ∈ X, ε > 0, and A, B ⊂ X. We use the following notation:

• Bd(x, ε) =^{y ∈ X d(x, y) < ε^}— the ε-neighborhood of x in X

(or the open ball with center x and radius ε);

1 Their plurals are compacta and continua, respectively.

2 For the proof, refer to Preliminaries (p.2) of [GAGT].

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1.1 Terminology and Notation 3

• Bd(x, ε) =^{y ∈ X d(x, y) 6 ε^}— the closed ε-neighborhood of x in X (or the closed ball with center x and radius ε);

• Nd(A, ε) =^∪_x∈ABd(x, ε) — the ε-neighborhood of A in X;

• diamdA = sup^{d(x, y) x, y ∈ A^}— the diameter of A;

• d(x, A) = inf^{d(x, y) y ∈ A^}— the distance of x from A;

• distd(A, B) = inf^{d(x, y) x ∈ A, y ∈ B^}— the distance of A and B. It should be noticed that Nd({x}, ε) = Bd(x, ε) and d(x, A) = distd({x}, A). For a collection A of subsets of X, the mesh of A is deﬁned as follows:

• meshdA = sup^{diamdA A ∈ A^}.

If there is no possible confusion, we can drop the subscript d and write B(x, ε), B(x, ε), N(A, ε), diam A, dist(A, B) and mesh A.

The standard spaces are listed below:

• Rⁿ — the n-dimensional Euclidean space with the norm

∥x∥ =^√x(1)²+ · · · + x(n)²,

0 = (0, . . . , 0) ∈Rⁿ — the origin, the zero vector or the zero element, ei ∈_Rⁿ — the unit vector deﬁned by ei(i) = 1 and ei(j) = 0 for j ̸= i;

• Sⁿ⁻¹=^{x ∈_Rⁿ ∥x∥ = 1^}— the unit (n − 1)-sphere;

• Bⁿ=^{x ∈_Rⁿ ∥x∥ 6 1^}— the unit closed n-ball;

• ∆ⁿ=^{x ∈ (_R+)ⁿ⁺¹ ∑ⁿ⁺¹_i=1 x(i) = 1^}— the standard n-simplex;

• Q = [−1, 1]^N— the Hilbert cube;

• s =_R^N — the space of sequences;

• µ⁰=^{{ ∑}^∞_i=12xi/3ⁱ xi∈ {0, 1}^}— the Cantor (ternary) set;

• ν⁰=R \ Q — the space of irrationals;

• 2 = {0, 1} — the discrete space of two points.

Note that Sⁿ⁻¹, Bⁿ, and ∆ⁿ are not product spaces even though the same notations are used for product spaces, where the indexes n−1 and n represent their dimensions (the indexes of µ⁰and ν⁰ are identical).

As is well-known, the countable product 2^Nof the discrete space 2 = {0, 1} is homeomorphic to the Cantor set µ⁰. On the other hand, the countable product N^N of the discrete space N of natural numbers is homeomorphic to the space ν⁰ of irrationals.³

Let A and B be collections of subsets of X and Y ⊂ X. We deﬁne

• A ∧ B = {A ∩ B | A ∈ A, B ∈ B};

• A|Y = {A ∩ Y | A ∈ A};

• A[Y ] = {A ∈ A | A ∩ Y ̸= ∅}.

The star of Y with respect to A is deﬁned as follows:

• st(Y, A) = Y ∪^∪A[Y ] (= Y ∪^∪_{A∈A[Y ]}A).

3 For the proof, refer to Preliminaries (pp.3–5) in [GAGT].

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When each A ∈ A is contained in some B ∈ B, it is said that that A refines B and denoted by

A ≺ B or B ≻ A.

It is said that A covers Y (or A is a cover of Y in X) if Y ⊂^∪A (=^∪_A∈AA). When A is a cover of Y in X, st(Y, A) =^∪A[Y ]. A cover of X in X is a cover of X. A cover of Y in X is said to be open (resp. closed) in X depending on whether its members are open (resp. closed) in X. If A is an open cover of X, then A|Y is an open cover of Y and A[Y ] is an open cover of Y in X. When A and B are open covers of X, A ∧ B is also an open cover of X. For covers A and B of X, it is said that A is a refinement of B if A ≺ B, where A is an open (resp. closed) refinement if A is an open (resp. closed) cover. For a space X, we denote

• cov(X) — the collection of all open covers of X. For U , V ∈ cov(X), we deﬁne

st(U, V) = {st(U, V) | U ∈ U}.

In the case of V = U, st(U, U ) is denoted by st U. When st U ≺ V, we call U a star-refinement of V. We inductively deﬁne stⁿU, n ∈N, as follows:

stⁿU = st(stⁿ⁻¹U, U), where st⁰U = U (so st¹U = st U).

Let (Xγ)γ∈Γ be a family of (topological) spaces and X =^∪_γ∈ΓXγ. The weak topology on X with respect to (Xγ)γ∈Γ is the topology deﬁned as follows:

U ⊂ X is open in X ⇔ ∀γ ∈ Γ, U ∩ Xγ is open in Xγ

(

A ⊂ X is closed in X ⇔ ∀γ ∈ Γ, A ∩ Xγ is closed in Xγ

) .

Suppose that X has the weak topology with respect to (Xγ)γ∈Γ and the topologies of Xγ and Xγ^′ agree on Xγ∩ Xγ^′ for any γ, γ^′ ∈ Γ . If Xγ∩ Xγ^′ is closed (resp. open) in Xγ for any γ, γ^′∈ Γ , then each Xγis closed (resp. open) in X and the original topology of each Xγ is a subspace topology inherited from X. In case Xγ ∩ Xγ^′ = ∅ for γ ̸= γ^′, X is the topological sum of (Xγ)γ∈Γ and denoted by X =^⊕_γ∈ΓXγ.

Let f : A → Y be a map from a closed set A in a space X to another space Y . The adjunction space Y ∪fX is the quotient space (X ⊕ Y )/∼, where X ⊕ Y is the topological sum and ∼ is the equivalence relation corresponding to the decomposition of X ⊕ Y into singletons {x}, x ∈ X \ A, and sets {y} ∪ f⁻¹(y), y ∈ Y (the latter is a singleton {y} if y ∈ Y \ f (A)). In case Y is a singleton, Y ∪f X ≈ X/A. One should note that the adjunction spaces are not Hausdorff in general. It is necessary to require some conditions for the adjunction space to be Hausdorﬀ.

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1.1 Terminology and Notation 5 Let f : X → Y be a map. For A ⊂ X and B ⊂ Y , we denote

f (A) =^{f (x) x ∈ A^} and f⁻¹(B) =^{x ∈ X f(x) ∈ B^}. For collections A and B of subsets of X and Y , respectively, we denote

f (A) =^{f (A) A ∈ A^} and f⁻¹(B) =^{f⁻¹(B) B ∈ B^}.

The restriction of f to A ⊂ X is denoted by f |A. It is said that a map g : A → Y extends over X if there is a map f : X → Y such that f |A = g. Such a map f is called an extension of g.

Let [a, b] be a closed interval, where a < b (∈R). A map f : [a, b] → X is called a path (from f (a) to f (b)) in X, where it is said that two points f (a) and f (b) are connected by the path f in X. An embedding (i.e., an injective path) f : [a, b] → X is called an arc (from f (a) to f (b)) in X, and the image f ([a, b]) is also called an arc. Namely, a space is called an arc if it is homeomorphic to I. A space X is path-connected (or arcwise connected) if each pair of distinct points x, y ∈ X are connected by a path (or an arc). It is said that X is locally path-connected (or locally arcwise connected) if any neighborhood U of each point x ∈ X contains a neighborhood V of x such that each pair of distinct points in V are connected by a path (or an arc) in U (i.e., for each y, z ∈ V , there is a path (or an arc) f : I → U such that f (0) = y and f (1) = z). In this deﬁnition, as is easily observed, V may be a path-connected (or an arcwise connected). Although the (local) arcwise connectedness looks to be stronger than the (local) path-connectedness, it is known that they are the same concepts, that is, the following proposition

holds (cf. Corollary 5.14.7 in [GAGT]). [5.14.7]

Proposition 1.1.1 An arbitrary space X is path-connected if and only if X is arcwise connected. Moreover, X is locally path-connected if and only if X is locally arcwise connected. ⊓⊔

For spaces X and Y , we denote

• C(X, Y ) — the set of (continuous) maps from X to Y .

Given subspaces X1, . . . , Xn ⊂ X and Y1, . . . , Yn ⊂ Y , a map f : X → Y is said to be a map from (X, X1, . . . , Xn) to (Y, Y1, . . . , Yn) and written

f : (X, X1, . . . , Xn) → (Y, Y1, . . . , Yn) if f (X1) ⊂ Y1, . . . , f (Xn) ⊂ Yn. We denote

• C((X, X1, . . . , Xn), (Y, Y1, . . . , Yn))

— the set of maps from (X, X1, . . . , Xn) to (Y, Y1, . . . , Yn);

• C((X, x0), (Y, y0)) = C((X, {x0}), (Y, {y0})). For maps f, g : X → Y (i.e., f, g ∈ C(X, Y )),

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• f ≃ g means that f and g are homotopic (or f is homotopic to g), that is, there is a map h : X × I → Y such that h0 = f and h1 = g, where ht: X → Y , t ∈ I, are deﬁned by ht(x) = h(x, t), and h is called a homotopy from f to g (between f and g). When g is a constant map, it is said that f is null-homotopic and denoted by f ≃ 0. For a homotopy h : X × I → Y , we call h({x} × I), x ∈ X, the tracks of h, where each h({x} × I) is the track of x ∈ X by h.

For spaces X and Y ,

• X ≃ Y means that X and Y are homotopy equivalent,⁴

that is, there are maps f : X → Y and g : Y → X such that gf ≃ idX and f g ≃ idY, where f is called a homotopy equivalence and g is a homotopy inverse of f . For each f, f^′ ∈ C(X, Y ) and g, g^′ ∈ C(Y, Z), we have the following:

f ≃ f^′, g ≃ g^′⇒ gf ≃ g^′f^′.

A homotopy h between maps f, g ∈ C((X, X1, . . . , Xn), (Y, Y1, . . . , Yn)) requires the condition that ht∈ C((X, X1, . . . , Xn), (Y, Y1, . . . , Yn)) for every t ∈ I, that is, h is regareded as the map

h : (X × I, X1× I, . . . , Xn× I) → (Y, Y1, . . . , Yn). When there are maps

f : (X, X1, . . . , Xn) → (Y, Y1, . . . , Yn), g : (Y, Y1, . . . , Yn) → (X, X1, . . . , Xn) such that gf ≃ idX and f g ≃ idY, we denote

• (X, X1, . . . , Xn) ≃ (Y, Y1, . . . , Yn).

For A ⊂ X, a homotopy h : X × I → Y is called a homotopy relative to A if h({x} × I) is degenerate (i.e., a singleton) for every x ∈ A. When a homotopy from f to g is a homotopy relative to A (where f |A = g|A), we denote

• f ≃ g rel. A.

Let f, g : X → Y be maps and U a collection of subsets of Y (in usual, U ∈ cov(Y )). It is said that f and g are U-close (or f is U-close to g) if

{{f (x), g(x)} x ∈ X^}≺ U ∪^{{y} y ∈ Y^},

which implies that U covers the set {f (x), g(x) | f (x) ̸= g(x)}. A homotopy h is called a U -homotopy if the collection of non-degenerate tracks of h reﬁnes U, that is,

4 It is also said that X and Y have the same homotopy type or X has the homotopy typeof Y .

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1.2 Banach Spaces in the Product of Real Lines 7 {h({x} × I) x ∈ X^}≺ U ∪^{{y} y ∈ Y^}.

In this case, U covers the set

∪ {h({x} × I) h({x} × I) is non-degenerate^}.

It is said that f and g are U -homotopic (or f is U -homotopic to g) and denoted by f ≃Ug if there is a U-homotopy h : X × I → Y such that h0= f and h1= g.

When Y = (Y, d) is a metric space, we denote

d(f, g) = sup^{d(f (x), g(x)) x ∈ X^}.

In general, the case d(f, g) = ∞ might be possible. If Y is bounded or X is compact, this d is a metric on the set C(X, Y ), which is called the sup-metric. For ε > 0, it is said that f and g are ε-close or f is ε-close to g if d(f, g) < ε. A homotopy h is called an ε-homotopy if mesh{h({x} × I) | x ∈ X} < ε. It is said that f and g are ε-homotopic and denoted by f ≃ε g if there is an ε-homotopy h : X × I → Y such that h0= f and h1= g.

In the above, even if d is not a metric on C(X, Y ) (i.e., d(f, g) = ∞ for some f, g ∈ C(X, Y )), it induces a topology on C(X, Y ) with a neighborhood basis of each f consisting of the following:

Bd(f, ε) =^{g ∈ C(X, Y ) d(f, g) < ε^}, ε > 0. which is called the uniform convergence topology.

1.2 Banach Spaces in the Product of Real Lines

Throughout this section, let Γ be an inﬁnite set. Here, we review Banach spaces⁵ being linear subspaces of the productR^Γ. We denote

• Fin(Γ ) — the set of all non-empty ﬁnite subsets of Γ .

Then, note that card Fin(Γ ) = card Γ . The product spaceR^Γ is a linear space with the following scalar multiplication and addition:

R^Γ ^×R ∋ (x, t) 7→ tx = (tx(γ))γ∈Γ ∈_R^Γ; R^Γ ^×R^Γ ∋ (x, y) 7→ x + y = (x(γ) + y(γ))γ∈Γ ∈_R^Γ.

With respect to the product topology of_R^Γ, these operations are continuous. Namely, _R^Γ with the product topology is a topological linear space.⁶ Note that w(_R^Γ) = ℵ0card Fin(Γ ) = card Γ .

5 A complete normed linear space is called a Banach space.

6 _{Refer to}

§3.4 in [GAGT].

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For each γ ∈ Γ , we deﬁne the unit vector eγ ∈ _R^Γ by eγ(γ) = 1 and eγ(γ^′) = 0 for γ^′ ̸= γ. It should be noticed that {eγ | γ ∈ Γ } is not a Hamel basis for_R^Γ and its linear span⁷is the following:

R^Γf ⁼

{x ∈_R^Γ x(γ) = 0 except for ﬁnitely many γ ∈ Γ^},

which is a dense linear subspace of _R^Γ. The subspace _R^N_f of s = _R^N is also denoted by sf.

As is easily observed, the following are equivalent: (a) R^Γ is metrizable;

(b) R^Γf is metrizable; (c) _R^Γ_f is ﬁrst countable; (d) card Γ 6 ℵ0.

Thus, in case Γ is uncountable, every linear subspace L ofR^Γ withR^Γ_f ⊂ L is non-metrizable. Moreover,_R^Γ (or_R^Γ_f) is metrizable only when Γ is countable. In case card Γ = ℵ0,R^Γ is linearly homeomorphic to the space of sequences s=R^N, that is, there exists a linear homeomorphism betweenR^Γ and s. On the other hand, we have the following proposition:

[1.2.1] Proposition 1.2.1 Let Γ be an infinite set. Then, any norm on_R^Γ_f does not induce the topology inherited from the product topology of R^Γ. Consequently, every linear subspace L of R^Γ withR^Γf ⊂ L is not normable.

We can consider various norms deﬁned on linear subspaces ofR^Γ, which are not compatible with the product topology as in Proposition 1.2.1 above. In general, the unit closed ball and the unit sphere of a normed linear space X = (X, ∥ · ∥) are denoted by BX and SX respectively, that is,

BX =^{x ∈ X ∥x∥ 6 1^} and SX =^{x ∈ X ∥x∥ = 1^}.

The zero vector (the zero element) of X is denoted by 0X, or simply by 0 if there is no possible confusion.

The Banach space ℓ∞(Γ ) and its closed linear subspaces c(Γ ) ⊃ c0(Γ ) are deﬁned as follows:

• ℓ∞(Γ ) =^{x ∈_R^Γ sup_γ∈Γ|x(γ)| < ∞^}with the sup-norm

∥x∥∞= sup

γ∈Γ

|x(γ)|;

• c(Γ ) =^{x ∈_R^Γ ∃t ∈ R such that ∀ε > 0, |x(γ) − t| < ε

except for ﬁnitely many γ ∈ Γ^};

• c0(Γ ) =^{x ∈_R^Γ ∀ε > 0, |x(γ)| < ε except for ﬁnitely many γ ∈ Γ^}.

7 The linear subspace generated by a set B is called the linear span of B.

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1.2 Banach Spaces in the Product of Real Lines 9 These are linear subspaces ofR^Γ but not topological ones as saw in the above. The space c(Γ ) is linearly homeomorphic to c0(Γ ) ×R by the following correspondence:

c0(Γ ) ×R ∋ (x, t) 7→ (x(γ) + t)γ∈Γ ∈ c(Γ ).

This correspondence and its inverse are Lipschitz with respect to the norm

∥(x, t)∥ = max{∥x∥^∞^,|t|}.

Furthermore,R^Γf with this norm is denoted by ℓ^f_∞(Γ ). Then, ℓ^f_∞(Γ ) ⊂ c0(Γ ) ⊂ c(Γ ) ⊂ ℓ∞(Γ ).

For the weight of these spaces, we have the following:

w(ℓ∞(Γ )) = 2^{card Γ} but w(c(Γ )) = w(c0(Γ )) = w(ℓ^f_∞(Γ )) = card Γ.⁸ The topology of ℓ^f_∞(Γ ) is diﬀerent from the topology inherited from the product topology. Indeed, {eγ | γ ∈ Γ } is discrete in ℓ^f_∞(Γ ) but 0 is a cluster point of this set with respect to the product topology.

In case Γ =N, we write

• ℓ∞(_{N) = ℓ}∞ — the space of bounded sequences,

• c(N) = c — the space of convergent sequences, and

• c0(_{N) = c}0 — the space of sequences convergent to 0.

We also write ℓ^f_∞(N) = ℓ^f∞, which is diﬀerent from sf as noted above. It should be noted that c and c0 are separable but ℓ∞ is non-separable. When card Γ = ℵ0, the spaces ℓ∞(Γ ), c(Γ ) and c0(Γ ) are linearly isometric to these spaces ℓ∞, c and c0, respectively.

Here, we regard Fin(Γ ) as a directed set by ⊂. For x ∈R^Γ, we say that

∑

γ∈Γx(γ) is convergent if^{( ∑}_γ∈Fx(γ)⁾_{F ∈Fin(Γ )}is convergent and deﬁne

∑

γ∈Γ

x(γ) = lim

F ∈Fin(Γ )

∑

γ∈F

x(γ).

In case x(γ) > 0 for all γ ∈ Γ , ^∑_γ∈Γx(γ) is convergent if and only if ( ∑

γ∈F^x(γ)

)

F ∈Fin(Γ ) is upper bounded, and then

∑

γ∈Γ

x(γ) = sup

F ∈Fin(Γ )

∑

γ∈F

x(γ).

Thus, by^∑_γ∈Γx(γ) < ∞, we mean that^∑_γ∈Γx(γ) is convergent.

For x ∈_R^N,^∑_i∈Nx(i) should be distinguished from^∑^∞_i=1x(i). When the sequence^{( ∑}ⁿ_i=1x(i)⁾_n∈Nis convergent, we say that^∑^∞_i=1x(i) is convergent and deﬁne

8 Cf. Proposition 1.2.2 in [GAGT].

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∑∞ i=1

x(i) = lim

n→∞

∑n i=1

x(i).

Evidently, if ^∑_i∈Nx(i) is convergent then ^∑^∞_i=1x(i) is also convergent and

∑∞

i=1^{x(i) =}

∑

i∈Nx(i). However,^∑_i∈Nx(i) is not convergent even if^∑^∞_i=1x(i) is convergent. In fact, due to Proposition 1.2.2 below,

∑

i∈N

x(i) is convergent ⇔

∑∞ i=1

|x(i)| is convergent.

[1.2.3] Proposition 1.2.2 For an infinite set Γ and x ∈R^Γ,^∑_γ∈Γx(γ) is convergent if and only if ^∑_γ∈Γ|x(γ)| < ∞. In this case, Γx= {γ ∈ Γ | x(γ) ̸= 0} is countable and ^∑_γ∈Γx(γ) =^∑^∞_i=1x(γi) for any sequence (γi)i∈N in Γ such that Γx⊂ {γi| i ∈_{N} and γ}i̸= γj if i ̸= j.

For each p > 1, the Banach space ℓp(Γ ) is deﬁned as follows:

• ℓp(Γ ) =^{x ∈_R^Γ ∑_γ∈Γ|x(γ)|^p< ∞^}with the norm

∥x∥p=

(∑

γ∈Γ

|x(γ)|^p )1/p

.

Similar to ℓ^f_∞(Γ ), the space_R^Γ_f with this norm is denoted by ℓ^f_p(Γ ).⁹ In the same way as c0(Γ ), we can show w(ℓp(Γ )) = card Γ . When card Γ = ℵ0, the Banach space ℓp(Γ ) is linearly isometric to ℓp = ℓp(N), which is separable. The space ℓ2(Γ ) is the Hilbert space with the inner product

⟨x, y⟩ = ^∑

γ∈Γ

x(γ)y(γ),

which is well-deﬁned because

∑

γ∈Γ

|x(γ)y(γ)| 6 ¹₂(∥x∥²₂+ ∥y∥²₂) < ∞.

For 1 6 p < q, we have ℓp(Γ )_{$ ℓ}q(Γ )_{$ c}0(Γ ) as sets (or linear spaces). These inclusions are continuous because ∥x∥∞ ⁶∥x∥q ⁶∥x∥p for every x ∈ ℓp(Γ ). When Γ is inﬁnite, the topology of ℓp(Γ ) is distinct from the one induced by the norm ∥ · ∥q or ∥ · ∥∞ (i.e., the topology inherited from ℓq(Γ ) or c0(Γ )). In fact, the unit sphere Sℓp(Γ ) is closed in ℓp(Γ ) but not closed in ℓq(Γ ) for any q > p nor in c0(Γ ).¹⁰

For 1 6 p 6 ∞, we haveR^Γ_f ⊂ ℓp(Γ ) as sets (or linear spaces). By ℓ^f_p(Γ ), we denote the subspace of ℓp(Γ ) with ℓ^f_p(Γ ) =_R^Γ_f as sets. When Γ =_{N, we}

9 The triangle inequality for_∥x∥pis known as the Minkowski inequality. The proof can be found in pp.16–17 in [GAGT].

10See p.17 in [GAGT].

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1.2 Banach Spaces in the Product of Real Lines 11 write ℓ^f_p(N) = ℓ^fp. By Proposition 1.2.1, we know ℓ^f_p(Γ ) ̸= R^Γf as spaces for any inﬁnite set Γ . In the above, the sequence (xn)n∈Nis contained in the unit sphere S_ℓ^f

p(Γ ) ^{of ℓ} f

p(Γ ), which means that S_ℓ^f

p(Γ ) is not closed in ℓ^f_q, hence ℓ^f_p ̸= ℓ^f_q as spaces for 1 6 p < q 6 ∞. Note that S_ℓ^f_p_{(Γ )} is a closed subset of ℓ^f_q for 1 6 q < p.

Concerning the convergence of sequences in ℓp(Γ ), we have the following:

[1.2.4] Proposition 1.2.3 For each p ∈N and x ∈ ℓp(Γ ), a sequence (xn)n∈N con-

verges to x in ℓp(Γ ) if and only if

∥x∥p= lim

n→∞^∥xⁿ^∥^p and x(γ) = lim

n→∞^xⁿ(γ) for every γ ∈ Γ .

Remark 1.1 It should be noted that Proposition 1.2.3 is valid for not only sequences but also nets, which means that the unit spheres Sℓp(Γ ), p ∈_{N, are} subspaces of the product spaceR^Γ, whereasR^Γ norR^Γf are not metrizable if Γ is uncountable. Therefore, if 1 6 p < q 6 ∞, then Sℓp(Γ ) is also a subspace of ℓq(Γ ), while, as mentioned above, Sℓp(Γ ) of ℓp(Γ ) is not closed in the space ℓq(Γ ). The unit sphere S_ℓ^f

p(Γ ) ^{of ℓ}

fp(Γ ) is a subspace of _R^Γ_f (⊂_R^Γ) and also a subspace of ℓq(Γ ) for 1 6 q 6 ∞.

Remark 1.2 The “if” part of Proposition 1.2.3 does not hold for the space c0(Γ ) for any inﬁnite set Γ (but the “only if” part obviously does hold).¹¹

Concerning the topological classiﬁcation of ℓp(Γ ), we have the following theorem due to S. Mazur:

[1.2.5] Theorem 1.2.4 (Mazur) For each 1 < p < ∞, ℓp(Γ ) is homeomorphic to

ℓ1(Γ ). By the same homeomorphism, ℓ^f_p(Γ ) is also homeomorphic to ℓ^f₁(Γ ). Proof. We deﬁne φ : ℓ1(Γ )_{→ ℓ}p(Γ ) and ψ : ℓp(Γ )_{→ ℓ}1(Γ ) as follows:

φ(x)(γ) = sign x(γ)_{· |x(γ)|}^1/p for x_{∈ ℓ}1(Γ ), ψ(x)(γ) = sign x(γ)_{· |x(γ)|}^p for x_{∈ ℓ}p(Γ ),

where sign 0 = 0 and sign a = a/|a| for a ̸= 0. We can apply Proposition 1.2.3 to verify the continuity of φ and ψ. In fact, the following functions are continuous:

ℓ1(Γ )∋ x 7→ ∥φ(x)∥^p⁼⁽∥x∥¹⁾^1/p∈ R, ℓ¹^{(Γ )}∋ x 7→ φ(x)(γ) ∈ R, γ ∈ Γ ; ℓp(Γ )∋ x 7→ ∥ψ(x)∥¹⁼⁽∥x∥^p⁾^p∈ R, ℓ^p^{(Γ )}∋ x 7→ ψ(x)(γ) ∈ R, γ ∈ Γ. Observe that ψφ = id and φψ = id. Then, φ is a homeomorphism with φ⁻¹= ψ, where φ(ℓ^f_p(Γ ))_{⊂ ℓ}^f₁(Γ ) and ψ(ℓ^f₁(Γ ))_{⊂ ℓ}^f_p(Γ ). _⊓_⊔

11Cf. Remark 3 in p.17 of [GAGT].

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For each space X, we denote C(X) = C(X,R). The Banach space C^B(X) is deﬁned as follows:

• C^B(X) =^{f ∈ C(X) supx∈X|f (x)| < ∞^}with the sup-norm

∥f ∥ = sup

x∈X

|f (x)|.

This sup-norm of C^B(X) induces the uniform convergence topology. If X is discrete and inﬁnite, then we have C^B(X) = ℓ∞(X), so C^B(_{N) = ℓ}∞ in particular. When X is compact, C^B(X) = C(X) and the topology induced by the norm coincides with the compact-open topology.

The uniform convergence topology of C(X) is induced by the metric deﬁned as follows:

d(f, g) = sup

x∈X

min{|f (x) − g(x)|, 1}.

As is easily observed, C^B(X) is a closed and open subspace of the space C(X) with the uniform convergence topology. Note that C^B(X) is a component of the space C(X) because C^B(X) is path-connected as a normed linear space.

Regarding C(X) as a subspace of the product spaceR^X, we can introduce a topology on C(X), which is called the pointwise convergence topology. With respect to this topology,

n→∞lim ^fⁿ ^{= f} ^⇔ n→∞^lim ^fⁿ(x) = f (x) for every x ∈ X.

The space C(X) with the pointwise convergence topology is usually denoted by Cp(X). The space Cp(N) is none other than the space of sequences s = R^N^. Among three topologies on C(X) considered above, the compact-open topology, the uniform convergence topology, and the pointwise convergence topology, the uniform convergence topology is the ﬁnest and the pointwise convergence topology is the coarsest.

1.3 Topological Spaces

A perfect map f : X → Y is a closed map such that f⁻¹(y) is compact for each y ∈ Y . A map f : X → Y is said to be proper if f⁻¹(K) is compact for every compact set K ⊂ Y . In the case that X and Y are metrizable, these concepts are coincide, that is,

[2.1.6] Proposition 1.3.1 For a map f : X → Y between metrizable spaces, the following are equivalent:

(a) f : X → Y is perfect; (b) f : X → Y is proper;

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1.3 Topological Spaces 13 (c) Any sequence (xn)n∈N in X has a convergent subsequence if (f (xn))n∈N

is convergent in Y .

This can be proved as follows:

(a)⇒ (b): Let K ⊂ Y be compact and U be an open cover of f⁻¹^{(K) in} X. By (a), for each y∈ K, we can choose a finite subcollection U^y ôf U so that f⁻¹(y)_⊂^∪_Uy. Since f is closed, each Vy= Y _{\ f(X \}^∪_Uy) is an open neighborhood of y in Y . Hence, there are y1, . . . , yn ∈ K such that K_⊂^∪ⁿ_i=1Vyi. Then, the finite subcollection^∪ⁿ_i=1_Uyi ⊂ U covers f⁻¹^(K). (b)⇒ (c): Let y = lim^n→∞^{f (x}ⁿ⁾∈ Y . Since K = {f(xⁿ⁾| n ∈ N} ∪ {y} is compact, so is f⁻¹(K) due to (b). Hence, the sequence (xn)n∈N has a convergent subsequence.

(c)⇒ (a): For each y ∈ Y , every sequence (xⁿ⁾^n∈N^{in f}⁻¹(y) has a convergent subsequence due to (c), which means that f⁻¹(y) is compact. For each closed set A⊂ X and y ∈ cl^Yf (A), take a sequence (xn)n∈Nin A so that y = limn→∞f (xn). Due to (c), (xn)n∈Nhas a convergent subsequence (xni⁾i∈N, i.e., we have limi→∞xni ^{= x} ∈ A. Then, y = f(x) ∈ f(A). Therefore, f (A) is closed in Y . _⊓_⊔

The following should be remarked:

[2.9.6]

• Let f : X → Y be a proper map and U be a locally ﬁnite open cover of Y such that every U ∈ U has the compact closure in Y (so X and Y should be locally compact). If a map g : X → Y is U -close to f , then g is also proper.

[2.2.2] Theorem 1.3.2 (Tietze Extension Theorem) Let A be a closed set in a

normal space X. Then, every map f : A → I extends over X.

Let A be a collection of subsets of a space X. It is said that A is locally finite (resp. discrete) in X if each point has a neighborhood U in X which meets only ﬁnitely many members (resp. at most one member) of A, that is, card A[U ] < ℵ0 (resp. card A[U ] 6 1). Moreover, A is σ-locally finite (resp. σ-discrete) in X if A is a countable union of locally ﬁnite (resp. discrete) subcollections.

[2.3.1] Theorem 1.3.3 (A.H. Stone) Every open cover of a metrizable space has

a locally finite and σ-discrete open refinement.

[2.3.4] Theorem 1.3.4 (Bing; Nagata–Smirnov) For a regular space X, the fol-

lowing conditions are equivalent: (a) X is metrizable;

(b) X has a σ-discrete open basis; (c) X has a σ-locally finite open basis.

The equivalence of (a) and (b) in the above theorem is called the Bing Metrization Theorem and the equivalence of (a) and (c) is called the Nagata–Smirnov Metrization Theorem. Separable metrizable spaces are characterized as follows:

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[2.3.5] Theorem 1.3.5 (Urysohn Metrization Theorem) A space is separable metrizable if and only if it is regular and second countable.

The (metric) completion of a metric space X = (X, d) is a complete metric space eX = ( eX, ˜d) containing X as a dense set and as a metric subspace (i.e., d is a restriction of ˜d).

[2.3.10] Theorem 1.3.6 Every metric space has a completion. ⊓⊔

A space¹²X is paracompact if each open cover of X has a locally ﬁnite open reﬁnement. By the A.H. Stone Theorem 1.3.3, every metrizable space is paracompact. We have the following characterization:

[2.6.3] Theorem 1.3.7 A space X is paracompact if and only if every open cover of X has an open star-refinement.

A space X is collectionwise normal if, for each discrete collection F of closed sets in X, there is a pairwise disjoint collection {UF | F ∈ F} of open sets in X such that F ⊂ UF for each F ∈ F. Obviously, every collectionwise normal space is normal. In the deﬁnition of collectionwise normality, {UF | F ∈ F} can be discrete in X. Indeed, choose an open set V in X so that

∪F ⊂ V ⊂ cl V ⊂ ^∪_{F ∈F}UF. Then, F ⊂ V ∩ UF for each F ∈ F and {V ∩ UF | F ∈ F} is discrete in X. A space X is hereditarily paracompact if every subspace of X is paracompact.

Proposition 1.3.8 (Results on Paracompact Spaces) [2.6.1] (1) Every paracompact space is collectionwise normal.

[2.6.7(1)] (2) A space is paracompact if it is a locally finite union of paracompact closed subspaces.

[2.6.7(2)] (3) Every Fσ subspace of a paracompact space is paracompact.

[2.6.7(3)] (4) A space X is hereditarily paracompact if every open subspace of X is paracompact.

[2.6.7(4)] (5) Every locally (completely) metrizable paracompact space is (completely) metrizable.

A closed set A ⊂ X is called a zero set in X if A = f⁻¹(0) for some map f : X →R. The complement of a zero set in X is called a cozero set. A normal space X is perfectly normal if every closed set in X is Gδ in X (i.e., every open set in X is Fσ in X), which is equivalent to the condition that every closed set in X is a zero set in X (i.e., every open set in X is a cozero set in X).¹³

Theorem 1.3.9 Every perfectly normal paracompact space is hereditarily paracompact. ⊓⊔

[2.6.8]

12Recall that spaces are assumed to be Hausdorff.

13Cf. Theorem 2.2.6 in [GAGT]

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1.3 Topological Spaces 15 It is said that a real-valued function f : X →R is lower semi-continuous (abbreviated as l.s.c.) (or upper semi-continuous (abbreviated as u.s.c.)) if f⁻¹((t, ∞)) (or f⁻¹((−∞, t))) is open in X for each t ∈R. Then, f : X → R is continuous if and only if f is l.s.c. and u.s.c.

[2.7.6] Theorem 1.3.10 Let g, h : X →R be real-valued functions on a paracompact

space X such that g is u.s.c., h is l.s.c. and g(x) < h(x) for each x ∈ X. Then, there exists a map f : X →R such that g(x) < f(x) < h(x) for each x ∈ X. Moreover, given a map f0 : A →R of a closed set A in X such that g(x) < f0(x) < h(x) for each x ∈ A, the map f can be an extension of f0.

A space X has the Baire property or is a Baire space if the intersection of countably many dense open sets in X is also dense, equivalently every countable intersection of dense Gδ sets in X is also dense. The Baire property can also be expressed as follows: If a countable union of closed sets has an interior point, then at least one of the closed sets has an interior point. The following statement is easily proved:

• Every open subspace and every dense Gδ subspace of a Baire space are also Baire.

[2.5.1] Theorem 1.3.11 (Baire Category Theorem) Every completely metriz-

able space is a Baire space. Consequently, it cannot be written as a union of countably many closed sets without interior points.

A metrizable space X is said to be absolutely Gδif X is Gδin an arbitrary metrizable space which contains X as a subspace. This concept characterizes the complete metrizability, that is,

[2.5.2] Theorem 1.3.12 A metrizable space is completely metrizable if and only if

it is absolutely Gδ.

For the complete metrizability, the following holds:

[2.5.3] Theorem 1.3.13 Let X = (X, d) be a complete metric space and A ⊂ X.

(1) If A is completely metrizable, then A is Gδ in X.

(2) If X is complete and A is Gδ in X, then A is completely metrizable.

[2.5.7] Theorem 1.3.14 (Lavrentieff) Let f : A → Y be a map from a subset A

of a space X to a completely metrizable space Y . Then, f extends over a Gδ

set G in X such that A ⊂ G ⊂ cl A.

[2.5.8] Theorem 1.3.15 (Lavrentieff) Let X and Y be completely metrizable

spaces and f : A → B be a homeomorphism between A ⊂ X and B ⊂ Y . Then, f extends to a homeomorphism ˜f : G → H between Gδ sets in X and Y such that A ⊂ G ⊂ cl A and B ⊂ H ⊂ cl B.

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Let P be a property for subsets of a space X. It is said that X has property P locally if each x ∈ X has a neighborhood U in X which has property P. A property P for open sets in X is said to be G-hereditary if the following conditions are satisﬁed:

(G-1) If U has property P, then every open subset of U has P; (G-2) If U and V have property P, then U ∪ V has property P;

(G-3) If {U_∪ λ | λ ∈ Λ} is discrete in X and each Uλ has property P, then

λ∈Λ^U^λ has property P.

The following theorem is very useful to show that a space has a certain property:

[2.6.5] Theorem 1.3.16 (E. Michael) Let P be a G-hereditary property for open sets in a paracompact space X. If X has property P locally then X itself has property P.

We can apply the above theorem even to a property P for closed sets in X by introducing the property P^◦as follow:

an open set U has property P^◦ ⇔

def cl U has property P It is said that P is F -hereditary if it satisﬁes the following conditions:

(F-1) If A has property P, then every closed subset of A has property P; (F-2) If A and B have property P, then A ∪ B has property P;

(F-3) If {A_∪ λ | λ ∈ Λ} is discrete in X and each Aλ has property P, then

λ∈Λ^A^λ has property P.

Evidently, if property P is F -hereditary, then P^◦ is G-hereditary. Therefore, Theorem 1.3.16 yields the following corollary:

[2.6.6] Corollary 1.3.17 (E. Michael) Let P be an F -hereditary property for closed sets in a paracompact space X. If X has property P locally, then X itself has property P. ⊓⊔

[2.7.7] Proposition 1.3.18 (Refinements by Open Balls)

(1) Let X be a metrizable space and U be an open cover of X. Then, X has an admissible bounded metric ρ such that

{Bρ(x, 1) x ∈ X^}≺ U.

Moreover, for given d ∈ Metr(X), ρ can be chosen so that ρ > d, which means that if d is complete then ρ is.

(2) Let X = (X, d) be a metric space. For each open cover U of X, there is a map γ : X → (0, 1) such that

{B(x, γ(x)) x ∈ X^}≺ U.

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1.4 Topological Linear Spaces 17

1.4 Topological Linear Spaces

Let E be a linear space.¹⁴ It is said that ﬁnitely many distinct points x1, . . . , xn∈ E are affinely (or geometrically) independent provided that, for t1, . . . , tn∈_R,

∑n i=1

tixi= 0,

∑n i=1

ti= 0 ⇒ t1= · · · = tn= 0,

that is, x1− xn, . . . , xn−1− xn are linearly independent. In this case, the set {x1, . . . , xn} ⊂ E is said to be affinely (or geometrically) independent. An (infinite) set A ⊂ E is said to be affinely (or geometrically) independent if every finite subset of A is affinely independent.

We call F ⊂ E a flat¹⁵ (resp. a convex set) if the straight line trough (resp. the line segment between) every distinct two points of F is contained in F . It is easy to see that F ⊂ E is a ﬂat if and only if F − x is a linear subspace of E for some (or any) x ∈ F , where F − x = F − y for any x, y ∈ F

(cf. [GAGT, Proposition 3.1.1, Corollary 3.1.2]). The dimension dim F of [3.1.1] [3.1.2] a ﬂat F is deﬁned as the dimension of the linear subspace F − x for some

(or any) x ∈ F . In the infinite-dimensional case, dim F is the cardinal of a maximal affinely independent subset. For a subset A ⊂ E, the smallest flat fl A is called the flat hull¹⁶ of A and the smallest convex set ⟨A⟩ is called the convex hull of A. The dimension dim C of a convex set C is defined as the dimension of fl C.

Let F and F^′be ﬂats (resp. convex sets) in linear spaces E and E^′, respectively. A function f : F → F^′ is said to be affine (or linear in the affine sense) provided that

f ((1 − t)x + ty) = (1 − t)f (x) + tf (y)

for every x, y ∈ F and t ∈R (resp. t ∈ I). A topological linear space E is a linear space with a topology such that the algebraic operations, the addition (x, y) 7→ x + y and the scalar multiplication (t, x) 7→ tx, are continuous.¹⁷

[3.5.1] Proposition 1.4.1 Every finite-dimensional flat F in an arbitrary linear

space E has the unique topology such that the following operation is continuous:

F × F ×R ∋ (x, y, t) 7→ (1 − t)x + ty ∈ F.

14Here, we only consider linear spaces over_R.

15A ﬂat is also called an affine set, a linear manifold, or a linear variety.

16The ﬂat hull is also called the affine hull.

17Recall that topological spaces are assumed to be Hausdorff. For topological linear spaces (more generally for topological groups), it suﬃces to assume the axiom T0, which implies the regularity (cf. Proposition 2.4.2 in [GAGT]).

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With respect to this topology, every affine bijection f : Rⁿ → F is a homeomorphism, where n = dim F .

As a corollary, we have the following:

[3.5.2] Corollary 1.4.2 Every finite-dimensional flat F in an arbitrary linear space E has the unique topology compatible with the algebraic operations (addition and scalar multiplication), and then it is linearly homeomorphic to_Rⁿ, where n = dim E.

[3.4.15] Theorem 1.4.3 Every open convex set V in a topological linear space E is homeomorphic to E itself.

For topological linear spaces, the ﬁnite-dimensionality can be topologically characterized as follows:

[3.5.9] Theorem 1.4.4 Let E be a topological linear space. The following are equivalent:

(a) E is finite-dimensional; (b) E is locally compact;

(c) 0 ∈ E has a totally bounded neighborhood in E.

The metrizability of a topological linear space has the following very simple characterization:

[3.6.1] Theorem 1.4.5 A topological linear space E is metrizable if and only if 0 ∈ E has a countable neighborhood basis.

A metric d on a linear space E is said to be invariant if d(x + z, y + z) = d(x, y) for every x, y, z ∈ E.

It is equivalent to this condition that d(x, y) = d(x − y, 0) for every x, y ∈ E. With respect to an invariant metric on E, addition on E is clearly continuous. Moreover, scalar multiplication on E is continuous if and only if the following three conditions are satisﬁed:

(i) d(xn, 0) → 0 ⇒ ∀t ∈_{R, d(tx}n, 0) → 0; (ii) tn→ 0 ⇒ ∀x ∈ E, d(tnx, 0) → 0;

(iii) d(xn, 0) → 0, tn→ 0 ⇒ ∀t ∈_{R, d(t}nxn, 0) → 0;

An invariant metric d on E satisfying these conditions is called a linear metric. A metric linear space E = (E, d) is a linear space E with a linear metric d. Evidently, every linear metric on a linear space E induces a topology of E, which makes E a topological linear space. Conversely, the following fact holds:

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1.4 Topological Linear Spaces 19

• Every admissible invariant metric for a metrizable topological linear space is a linear metric.

Refer to [GAGT, pp.111–112].

A functional ∥ · ∥ : E →R on a linear space E is called an F -norm if it satisﬁes the following conditions:

(F1) ∥x∥ > 0;

(F2) ∥x∥ = 0 ⇒ x = 0; (F3) |t| 6 1 ⇒ ∥tx∥ 6 ∥x∥; (F4) ∥x + y∥ 6 ∥x∥ + ∥y∥; (F5) ∥xn∥ → 0 ⇒ ∥txn∥ → 0; (F6) tn → 0 ⇒ ∥tnx∥ → 0.

It should be observed that the converse of (F2) is true because ∥0∥ = 0 by the last condition (F6). Then, ∥x∥ = 0 if and only if x = 0. A linear space E given an F -norm ∥ · ∥ is called an F -normed linear space. Every norm is an F -norm, hence every normed linear space is an F -normed space. An F -norm

∥ · ∥ induces the linear metric d(x, y) = ∥x − y∥. Then, every F -normed linear space is a metric linear space. An F -norm on a topological linear space E is said to be admissible if it induces the topology for E.

Theorem 1.4.6 A topological linear space has an admissible F -norm if and [3.6.3] only if it is metrizable.

For each metrizable topological linear space, there exists an F -norm with the following stronger condition than (F3):

(F₃^∗) x ̸= 0, |t| < 1 ⇒ ∥tx∥ < ∥x∥,

which implies that ∥sx∥ < ∥tx∥ for each x ̸= 0 and 0 < s < t.

A topological linear space E is locally convex if 0 ∈ E has a neighborhood basis consisting of (open) convex sets, equivalently open convex sets make up an open basis for E.

[3.8.9] Theorem 1.4.7 (Bartle–Graves–Michael) Let E be a locally convex

metric linear space and F be a linear subspace of E that is complete (so a Fr´echet space). Then, E ≈ F × E/F . In particular, E ≈ R × G for some metric linear space G.

[3.8.11] Theorem 1.4.8 (Banach–Mazur, Klee) For every Banach space E, there

is continuous linear surjection q : ℓ1(Γ ) → E, where card Γ = dens E. As a corollary of the above two theorem, we have the following:

[3.8.12] Corollary 1.4.9 For any Banach space E, there exists a Banach space F

such that E × F ≈ ℓ1(Γ ), where card Γ = dens E. ⊓⊔