A note on lattice renormings

A note on lattice renormings

Mari´an Fabian^∗, Petr H´ajek, V´aclav Zizler

Abstract. It is shown that every strongly lattice norm onc0(Γ) can be approximated by C^∞ smooth norms. We also show that there is no lattice and Gˆateaux differentiable norm onC0[0, ω1].

Keywords: smooth norms, approximation, lattice norms,c0(Γ),C0[0, ω1] Classification: 46B03, 46B20

It has been recently shown in [1] and [2] that every equivalent norm on the classical separable Banach spaces c₀ or ℓ_p, p even, (as well as on many other spaces) can be uniformly approximated on bounded sets by a sequence of C^∞- Fr´echet smooth norms.

Although the method of construction requires some technical conditions on the space to be satisfied (in particular the existence of a Schauder basis), it seems to suggest that perhaps the following statement should be valid:

SupposeX is a separable Banach space that admits an equivalentC^k-Fr´echet smooth norm. Then every equivalent norm onX can be approximated uniformly on bounded sets by a sequence ofC^k-Fr´echet smooth norms.

On the other hand, we do not know of any example of a nonseparable Banach space where a similar statement would be valid fork≥2.

In the present note we give a partial solution to this problem for the spacec₀(Γ) andk=∞. More precisely we show that onc₀(Γ), Γ uncountable, every equiv- alent strongly lattice norm can be approximated by a sequence of C^∞-Fr´echet smooth norms.

In the second part of our paper, we show that there exists no lattice Gˆateaux differentiable norm onC₀([0, ω₁]), the space of continuous functions on the ordi- nal segment [0, ω₁] that vanish atω₁ (where ω₁ is the first uncountable ordinal and [0, ω₁] is in its normal topology as in [4]). More information on the space C₀([0, ω₁]) can be found e.g. [3, p. 259]. Proposition 2 of this paper is of interest when compared with some results of Haydon [5]–[6]. In [5], a lattice norm on C₀[0, ω1]⊕c₀[0, ω1] is constructed, which isC^∞-Fr´echet differentiable and locally dependent on finitely many coordinates when restricted to a rather large open

In part supported by NSERC (Canada).

∗Supported by Grants AV 101–95–02 and GA ˇCR 201–94–0069

subset of C₀[0, ω₁]⊕c₀[0, ω₁]. This norm is then used to obtain C^∞-Fr´echet smooth (necessarily non-lattice) renormings ofC₀[0, ω₁].

The notation and terminology we use are mostly standard, as in [3].

By a strongly lattice norm onc₀(Γ) we mean an equivalent normk · ksuch that k P

γ∈Γ

yγeγk ≥ k P

γ∈Γ

xγeγkwhenever P

γ∈Γ

yγeγ, P

γ∈Γ

xγeγ∈c₀(Γ) are such that for everyγ∈Γ|yγ| ≥ |xγ|is satisfied.

Theorem 1. Every equivalent strongly lattice norm on c₀(Γ) can be approxi- mated(uniformly on bounded sets)byC^∞-Fr´echet smooth norms.

Proof: Denote the given strongly lattice norm byk · k. We first introduce an auxiliary functionf_∆. For arbitrary 1>∆>0 and P

γ∈Γ

x_γe_γ ∈c₀(Γ) denote by

f_∆ X

γ∈Γ

x_γe_γ

= sup

kX

γ∈Γ

y_γe_γk,

where y_γ=x_γ if |x_γ|>∆ and |y_γ| ≤∆ if |x_γ| ≤∆

. Clearly,f_∆(·)≥ k · k onc₀(Γ).

In fact,f_∆(·) is a Lipschitz function on (c₀(Γ),k · k∞) with the Lipschitz con- stant less than or equal to the Lipschitz constant ofk · k(on (c0(Γ),k · k∞)).

It is standard to check the following elementary properties off_∆(·):

(i) f_∆(P

γ∈Γ

x_γe_γ) = f_∆( P

γ∈{α∈Γ,|xα|>∆}

x_γe_γ). In other words, the value of f_∆(x) depends only on those coordinates ofxthat are in absolute value larger than ∆.

(ii) f_∆(P

γ∈Γ

x_γe_γ)≤f_∆(P

γ∈Γ

y_γe_γ)

whenever we havekyγk ≥ kxγkfor everyγ∈Γ.

The property (ii) is a “strongly lattice” property off_∆(·) and follows directly from the strongly lattice property ofk · k.

We now proceed with our construction of approximatingC^∞-norm.

Givenε >0, from the equivalence ofk · kandk · k_∞it follows that there exists 1>∆>0 such that

k · k ≤f_∆(·)≤ k · k+ε for everyx∈c₀(Γ).

PutF_∆(x) =f_∆²(x).

ThenF_∆(·) shares properties (i), (ii) and satisfies:

k · k²≤F_∆(·)≤(k · k+ε)²=k · k²+ 2εk · k+ε².

Thus the convex functionC_∆(·) defined by:

C_∆(x) = inf n

X

i=1

λ_iF_∆(x_i), x=

n

X

i=1

λ_ix_i,

n

X

i=1

λ_i= 1, λ_i>0

also satisfiesk·k²≤C_∆(·)≤(k·k+ε)², becausek·k²is convex andC_∆(·)≤F_∆9(·).

It is straightforward to show that also the strongly lattice property for C_∆(·) is preserved, i.e. C_∆(x)≥C_∆(y) forx, y ∈c₀(Γ), such that for everyγ∈Γ either kyγk ≥ kxγk. We will now show that for 1> ε >0 we have

C_∆(x) = inf n

X

i=1

λ_iF_∆(x_i), x=

n

X

i=1

λ_ix_i,

n

X

i=1

λ_i= 1, λ_i>0 andkx_ik ≤100

for everyx∈c₀(Γ) withkxk ≤2.

To this end, it is enough to find for every {x_i}ⁿ_i=1, λ_i > 0, Pⁿ

i=1

λ_i = 1, x = Pn

i=1

λ_ix_i another system{y_i}^m_i=1, λ^′_i >0, P^m

i=1

λ^′_i = 1,x= P^m

i=1

λ^′_iy_i, where ky_ik ≤ 100 and such that

m

X

i=1

λ^′_iF_∆(y_i)≤

n

X

i=1

λ_iF_∆(x_i).

Suppose without loss of generality thatkx_ik ≤100 for 1≤i≤jandkx_ik>100 for j < i ≤n. We may assume that j ≥1, since otherwise F_∆(x_i)≥ 100² for every 1≤i≤n, and thenF_∆(x)≤3²<100² would give us a better estimate.

Put

v₁=

j

P

i=1

λ_ix_i

j

P

i=1

λ_i

, v₂ = Pn i=j+1

λ_ix_i Pn i=j+1

λ_i ,

ξ₁=

j

X

i=1

λ_i, ξ₂= 1−ξ₁. Clearly,x=ξ₁v₁+ξ₂v₂.

We may assume thatF_∆(v₁)≥ _ξ¹₁

j

P

i=1

λ_iF_∆(x_i) and F_∆(v2)≥_ξ¹₂ Pⁿ

i=j+1

λ_iF_∆(x_i).

Indeed, if for example F_∆(v₁)< _ξ¹

1

j

P

i=1

λ_iF_∆(x_i), we obtain that x=ξ₁v₁+ Pn

i=j+1

λ_ix_i,ξ₁+ Pⁿ

i=j+1

λ_i = 1,ξ₁≥0,λ_i≥0 and

ξ₁F_∆(v₁) +

n

X

i=j+1

F_∆(x_i)<

n

X

i=1

λ_iF_∆(x_i) gives us even a better estimate ofC_∆(x).

By assumption, F_∆(x_i)≥100² forj+ 1≤i≤n. Thus _ξ¹

2

n

P

i=j+1

λ_iF_∆(x_i)≥ 100². The trivial estimate forC_∆(x) isF_∆(x)≤3²= 9. Thus _ξ¹

1

j

P

i=1

λ_iF_∆(x_i)≤ 9 (otherwise the trivial estimate would give us a smaller value than Pⁿ

i=1

λ_iF_∆(x_i) = ξ_{1 ξ}¹

1

j

P

i=1

λ_iF_∆(x_i) +ξ_{2 ξ}¹

2

Pn i=j+1

λ_iF_∆(x_i) .

Consequently,kv₁k² ≤C_∆(v₁)≤9 and we havekv₁k ≤3. Similarly, kv₂k+ ε2

≥F_∆(v₂)≥100² and we havekv₂k ≥99.

Thus there existsv₃∈c₀(Γ),kv₃k= 50,v₃=α₁v₁+α₂v₂ whereα₁+α₂= 1, α_i ≥0. Since v₃−α₁v₁=α₂v₂, we have 47≤α₂kv₂k. Thus

α₁ 1 ξ₁

j

X

i=1

λ_iF_∆(x_i) +α₂ 1 ξ₂

n

X

i=j+1

λ_iF_∆(x_i)≥α₂kv2k² ≥47kv2k ≥47·99.

Moreover the trivial estimate gives us

F_∆(v₃)≤ kv₃k+ε2≤51²<47·99.

Therefore

F_∆(v₃)≤α₁ 1 ξ₁

j

X

i=1

λ_iF_∆(x_i) +α₂ 1 ξ₂

n

X

i=j+1

λ_iF_∆(x_i), ξ₂

α₂F_∆(v₃)≤ ξ₂ α₂

α₁ ξ₁

j

X

i=1

λ_iF_∆(x_i) +

n

X

i=j+1

λ_iF_∆(x_i),

j

X

i=1

λ_iF_∆(x_i)− ξ₂ α₂

α₁ ξ₁

j

X

i=1

λ_iF_∆(x_i) + ξ₂

α₂F_∆(v₃)≤

n

X

i=1

λ_iF_∆(x_i),

j

X

i=1

1− ξ₂ α₂

α₁ ξ₁

λ_iF_∆(x_i) + ξ₂

α₂F_∆(v3)≤

n

X

i=1

λ_iF_∆(x_i).

However, 1− ξ₂

α₂ α₁

ξ₁

j

X

i=1

λ_ix_i+ ξ₂

α₂v₃=ξ₁v₁+ξ₂ v₃ α₂ −α₁

α₂v₁

=ξ₁v₁+ξ₂v₂=x.

It is easy to verify that

j

P

i=1

1−_α^ξ2₂^α_ξ¹

1

λ_i+_α^ξ2₂ = 1. It follows thatα₂> ξ₂, since kv3k= 50 whilekxk ≤2. Therefore (1−_α^ξ2₂^α_ξ1¹)λi ≥0 for every 1≤i≤j.

Thus the system{x_i}^j_i=1∪ {v₃},{ 1−_α^ξ2₂^α_ξ¹

1

λ_i}^j_i=1∪ {_α^ξ2₂}gives us a smaller estimate ofC_∆(x) than the original one {x_i}ⁿ_i=1, {λ_i}. Clearly, all kx_ik ≤ 100, 1≤i≤j,kv₃k ≤100.

Sincek ·kandk ·k∞are equivalent norms onc₀(Γ), it follows from our previous considerations that there exists a constantksuch that

C_∆(x) = inf ^j

X

i=1

λ_iF_∆(x_i), x=

j

X

i=1

λ_ix_i,

j

X

i=1

λ_i= 1, λ_i>0 andkx_ik_∞≤k

for everykxk ≤2.

We proceed by proving that there existsδ >0 such that

C_∆ X

γ∈Γ

x_γe_γ

=C_∆ X

γ∈{α,|xα|>δ}

x_γe_γ

for everyx= P

γ∈Γ

x_γe_γ∈c₀ such thatkxk ≤2.

In fact, we will show that choosingδ < _2k+2+∆^∆2 is sufficient.

SinceC_∆ is upper semi-continuous (as the infimum of a family of continuous functions - F_∆ is continuous as the square of a Lipschitz function f_∆), and, moreover, from the strongly lattice property ofC_∆it is enough to prove that

C_∆(X

γ∈Γ

x_γe_γ) =C_∆(X

γ∈Γ γ6=γ0

x_γe_γ),

whenever|xγ0| ≤δ.

We will proceed as follows. Givenx= P

γ∈Γ

x_γe_γ, for arbitrary{yi}ⁿ_i=1⊂c₀(Γ), {λ_i}ⁿ_i=1, λ_i > 0, Pⁿ

i=1

λ_i = 1, ky_ik_∞ ≤ k such that Pⁿ

i=1

λ_iy_i = P

γ∈Γ γ6=γ0

x_γe_γ, we will construct {x_i}ⁿ_i=1 ⊂c₀(Γ) such that (x_i)γ = (y_i)γ for 1 ≤ i ≤ n, γ 6= γ₀,

Pn i=1

λ_ix_i =xand in addition

n

X

i=1

λ_iF_∆(x_i)≤

n

X

i=1

λ_iF_∆(y_i).

Consequently,

C_∆(X

γ∈Γ

x_γe_γ)≤C_∆(X

γ∈Γ γ6=γ0

x_γe_γ).

This implies our claim, since C_∆(·) shares the strongly lattice property, so the opposite inequality is satisfied.

Without loss of generality assume that,δ≥xγ0 >0 and k≥(y_i)γ0 >∆ for 1≤i≤j₁,

∆≥(y_i)γ0 ≥0 for j₁ < i≤j₂, 0>(y_i)_γ0 ≥ −∆ for j₂ < i≤j₃,

−∆>(y_i)_γ0 ≥ −k for j₃ < i≤n.

Puts₁=

j1

P

i=1

λ_i,s₂=

j2

P

i=j1+1

λ_i,s₃ =

j3

P

i=j2+1

λ_i, s₄= Pⁿ

i=j3+1

λ_i. If (s₃+s₄)∆≥δ, then

j2

X

i=1

λ_i(y_i)γ0+

n

X

i=j2+1

λ_i∆≥

n

X

i=j2+1

λ_i∆≥(s₃+s₄)∆≥δ.

Therefore for everyj₂< i≤nwe can find numbers ˜y_i, such that ∆≥y˜_i≥(y_i)γ0

and j2

X

i=1

λ_i(y_i)γ0+

n

X

i=j2+1

λ_iy˜_i=x_γ0. We definex_i =y_i for 1≤i≤j₂, andx_i= P

γ∈Γ γ6=γ0

(y_i)γeγ+ ˜y_ieγ0 forj₂< i≤n. It follows that

F_∆(x_i) =F_∆ X

γ∈Γ γ6=γ0

(y_i)γeγ

≤F_∆(y_i).

Thus Pⁿ

i=1

λ_iF_∆(x_i)≤ Pⁿ

i=1

λ_iF_∆(y_i) and the claim is established.

If (s3+s₄)∆< δ, we obtain 0 = Pⁿ

i=1

λ_i(y_i)

γ0 ≥s₁∆−(s3+s₄)k. Therefore s₁ ≤ _∆^δk₂.Thuss₂ = 1−s₁−s₃−s₄≥1−^δ(k+1)_∆2 . We can find numbers ˜y_i for j₁< i≤j₂, such that (y_i)_γ0 ≤y˜_i ≤∆ and

j1

X

i=1

λ_i(y_i)_γ0+

n

X

i=j2+1

λ_i(y_i)_γ0+

j2

X

i=j1+1

λ_iy˜_i =x_γ0.

Indeed,

n

P

i=j2+1

λ_i(y_i)_γ0

≤ (s₃ +s₄)k ≤ ^δk_∆. Consequently, s₂∆− ^δk_∆ ≥ ∆−

δ(k+1)

∆ −^δk_∆ > δby our choice ofδ.

Putting (x_i)_γ = ˜y_i for j₁ < i ≤j₂, γ = γ₀ and (x_i)_γ = (y_i)_γ for any other choices ofiandγ, we obtain again

n

X

i=1

λ_iF_∆(x_i) =

n

X

i=1

λ_iF_∆(y_i).

Hence we proved thatC_∆(·) is a convex function on c₀(Γ), k · k² ≤C_∆(·) ≤ k · k+ε2

and, forkxk ≤2, C_∆(x) depends only on those coordinatesx_γ of x for which|xγ| ≥δ. More precisely,

C_∆ X

γ∈Γ

x_γe_γ

=C_∆ X

γ∈Γ1

x_γe_γ ,

where Γ1 ={γ∈Γ, |xγ| ≥δ}.

We will now construct aC^∞-Fr´echet smooth convex function on the set {x∈ c₀(Γ), kxk < 2}, which uniformly approximates C_∆(·). To this end, choose a C^∞-smooth bump function b(t) on R, 0 ≤ b(t) = b(−t), suppb ⊂ [−^δ₄,^δ₄],

∞

R

−∞

b(t)dt= 1.

It is elementary to check that from the symmetry condition on b and the convexity off it follows that

f(r)≤ Z∞

−∞

f(t)b(r−t)dt for arbitrary convex continuous function defined onR.

It is standard to check that for arbitraryγ₀∈Γ, the function C_∆^γ0 X

γ∈Γ

x_γe_γ

=

∞

Z

−∞

C_∆ X

γ∈Γ γ6=γ0

x_γe_γ+te_γ0

b(x_γ0−t)dt

is convex andC_∆^γ0(·)≥C_∆(·).

Put Π =

π={γ₁, . . . , γ_n}, n∈N, γ_i ∈Γ to be the set of all finite subsets of Γ. Forπ={γ₁, . . . , γ_n} ∈Π define

C_∆^π X

γ∈Γ

xγeγ

=

= Z∞

−∞

· · · Z∞

−∞

C_∆ X

γ∈Γ γ /∈π

x_γe_γ+

n

X

i=1

t_ie_γi

b(xγ1−t₁). . . b(xγn−t_n)dt₁. . . dt_n.

For every π ∈Π, C_∆^π is a convex function satisfying C_∆^π2(·)≥ C_∆^π1(·) whenever π₁ ⊂π₂.

Define ˜C_∆(x) = sup{C_∆^π(x), π∈Π}.

Supposex= P

γ∈Γ

x_γe_γ, kxk ≤ 2−^δ₂, Γ₁ ={γ ∈ Γ, |xγ| ≤ ^δ₄}, Γ₂ = Γ\Γ₁. Clearly Γ₂ ∈Π. For everyy∈c₀(Γ) such thatky−xk∞< ^δ₄, we have|yγ| ≤ ^δ₂ forγ∈Γ1. For suchy the following formula is satisfied:

C˜_∆(y) =C_∆^Γ2(y) =

= Z∞

−∞

· · · Z∞

−∞

C_∆ X

γ∈Γ γ /∈Γ2

y_γe_γ+

n

X

i=1

t_ie_γi

b(yγ1−t₁). . . b(yγn−t_n)dt₁. . . dt_n,

where Γ₂ ={γ₁, . . . , γ_n}.

Indeed, for every Γ₃={γ₁, . . . , γ_m}, Γ₂ ⊂Γ₃ we have

C_∆^Γ3(y) =

∞

Z

−∞

· · ·

∞

Z

−∞

C_∆ X

γ∈Γ γ /∈Γ3

y_γe_γ+

m

X

i=1

t_ie_γi

b(y_γ1−t₁). . . b(y_γn−t_n)dt₁. . . dt_m,

and thus C_∆^Γ3(y)

= Z∞

−∞

· · · Z∞

−∞

C_∆ X

γ∈Γ γ /∈Γ2

y_γe_γ+

n

X

i=1

t_ie_γi

b(yγ1 −t₁). . . b(yγn−t_n)dt₁. . . dt_n

=C^Γ2(y),

because the functionφ(t₁, . . . , t_m) =C_∆ P

γ∈Γ γ /∈Γ3

y_γe_γ+ P^m

i=1

t_ie_γi

is for any given

t₁, . . . , t_n constant in variables t_n+1, . . . , t_m satisfying |t_n+1−y_γn+1| ≤ ^δ₄, . . . ,

|tm−y_γm| ≤ ^δ₄. The function ˜C_∆(·) restricted toB_k·k∞ x,^δ₄

thus depends only on the coordinates {y_γ1, . . . , y_γn} of y and is easily observed to be C^∞-Fr´echet smooth. The trivial estimate gives us

kxk²≤C_∆(x)≤C˜_∆(x)≤sup{C_∆(x+v),kvk∞< δ 2}

≤sup{ kx+vk+ε2

, kvk_∞<δ 2}.

By the standard argument of choosing ε and δ small enough, we obtain, via the implicit function theorem, that theC^∞-Fr´echet smooth norm defined as the Minkowski functional of the set{x, C˜_∆(x)≤1}approximates arbitrary well (on bounded sets) the original normk · k.

We say that a normk| · k| defined on aC(K) space depends locally on finitely many coordinates if for every f ∈ C(K) there exist a finite set {k₁, . . . , k_n} ⊂ K, ε >0 andφ:Rⁿ→Rsuch that

|||g|||=φ g(k₁), . . . , g(k_n) ,

wheneverkg−fk< ε.

Proposition 2. There exists no lattice and Gˆateaux differentiable(not neces- sarily equivalent)normC₀([0, ω₁]). There exists no lattice(not necessarily equiv- alent)norm onC₀([0, ω₁])that depends locally on finitely many coordinates.

Proof: Assume thatk · kis a given norm onC₀([0, ω₁]). Let us first define, for a given non-limit ordinalα < ω₁,ϕ_α on [α, ω₁) by

ϕ_α(β) =kχ_[α,β]k forβ a nonlimit ordinal,

ϕ_α(β) = sup{ϕ_α(γ), γ < β, γ nonlimit}forβ a limit ordinal.

The functionϕ_αis well defined sinceχ_[α,β]∈C₀[0, ω₁] wheneverα, βare nonlimit ordinals. By the lattice condition onk · k,ϕ_α is a nondecreasing function defined on [0, ω₁). Thus for some nonlimitβ_α> α we have

ϕ_α(βα) =ϕ_α(γ) for everyγ∈[βα, ω₁].

Similarly, by the lattice assumption, whenever α₁ < α₂ are nonlimit ordinals, ϕ_α1(β_α1)≤ϕ_α2(β_α2). Therefore, there existsα₀∈ω₁ such that

ϕ_α0(βα0)≥ϕ_α(β) wheneverβ≥α≥α₀.

Let us define, by induction, a sequence{α_i}^∞_i=0 as follows: α₀ comes from the above consideration,α_i+1=β_αi+ 1.

Choose a closed and open countable interval [α₀, β]⊂[0, ω₁) such thatβ ≥α_i for everyi∈N. Clearly,χ_[α0,β]∈C₀([0, ω₁]) and

0<kχ_[α0,β]k=kχ_[αi,βαi]k for everyi∈N. Also,

kχ_[α0,β]+t χ_[αi,β

αi]k ≥ k(1 +t)χ_[αi,β

αi]k= (1 +t)kχ_[α0,β]k for everyt≥0.

Thus, the directional derivative of k · k at χ_[α0,β] in direction of v_i = χ_[αi,βαi] satisfies:

∂kχ_[α0,β]k

∂v_i ≥ ∂kχ_[αi,βαi]k

∂v_i ≥ kχ_[αi,βαi]k=kχ_[α0,β]k.

However, assuming the existence of the Gˆateaux derivativekχ_[α0,β]k^′, we estimate

kχ_[α0,β]k^′ ₁≥

hkχ_[α0,β]k^′,Pⁿ

i=0

v_ii Pn

i=0

v_i

= Pn i=0

∂kχ_[α0,β]k

∂vi

kχ_[α0,β]k ≥n

for all n∈ N. (kPⁿ

i=0

v_ik = kχ_[α0,β]k by the lattice property of k · k.) This is a

contradiction.

This proves the first half of Proposition 2. The proof for the second part requires only minor modifications.

Acknowledgment. We would like to thank the referee for suggesting some im- provements and for finding an error in the original version. The second author would also like to thank the Department of Mathematics, University of Alberta, for hospitality and support during the preparation of this note.

References

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[2] Deville R., Fonf V., H´ajek P.,Analytic andC^k-smooth approximations of norms in sepa- rable Banach spaces, Studia Math., to appear.

[3] Deville R., Godefroy G., Zizler V.,Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, 1993.

[4] Dugundji J.,Topology, Allyn and Bacon Inc., 1966.

[5] Haydon R.,Normes infiniment differentiables sur certains espaces de Banach, C.R. Acad.

Sci. Paris, t. 315, Serie I (1992), 1175–1178.

[6] Haydon R.,Trees in renormings theory, to appear.

Mathematical Institute, Czech Academy of Sciences, ˇZitn´a 25, 115 67 Prague 1, Czech Republic

E-mail: [email protected]

Mathematical Institute, Czech Academy of Sciences, ˇZitn´a 25, 115 67 Prague 1, Czech Republic

E-mail: [email protected]

Department of Mathematics, University of Alberta, Edmonton, T6G 2G1, Canada E-mail: [email protected]

(Received March 22, 1996)