Representation of bilinear forms in non-Archimedean Hilbert space by linear operators II

Representation of bilinear forms in non-Archimedean Hilbert space by linear operators II

Dodzi Attimu, Toka Diagana

Abstract. The paper considers the representation of non-degenerate bilinear forms on the non-Archimedean Hilbert spaceE_ω×E_ωby linear operators. More precisely, upon making some suitable assumptions we prove that ifϕis a non-degenerate bilinear form onE_ω×E_ω, thenϕis representable by a unique linear operatorAwhose adjoint operator A^∗exists.

Keywords: non-Archimedean Hilbert space, bilinear form, continuous linear functionals, non-Archimedean Riesz theorem, bounded bilinear form, stable unbounded bilinear form, unstable unbounded bilinear form

Classification: 47S10, 46S10

1. Introduction

Let K be a field, which is complete with respect to a non-Archimedean val- uation denoted | · |. Classical examples of such a field include Q_p, the field of p-adic numbers, wherep≥2 is a prime,C_p, the field ofp-adic complex numbers, and the field of Laurent series. Fix once and for all a sequence ω = (ω_i)_i∈N

of nonzero elements ofK and defineE_ω as the set of all sequences u= (u_i)_i∈N

of elements of K such that the series P

i∈Nω_iu²_i converges in K, equivalently, lim_i→∞(|u_i|.|ω_i|^1/2) = 0. A natural norm is defined onE_ω as follows:

u= (u_i)_i∈N, kuk= sup

i∈N

|u_i|.|ω_i|^1/2 . This norm is non-Archimedean in the sense that, for anyu, v∈E_ω,

ku+vk ≤max (kuk,kvk)

with the equality holding if kuk 6= kvk. An inner product (symmetric, bilin- ear, non-degenerate form) is defined on E_ω as follows: for all u = (u_i)_i∈N, v= (v_i)_i∈N∈E_ω,

hu, vi:=

∞

X

i=0

ω_iu_iv_i.

The vector spaceE_ωhas a special base, denoted (e_i)_i∈Nwheree_iis the sequence whose j-th term is 0 if i 6= j, and the i-th term is 1. This base satisfies the following: (i) for everyi, ke_ik=|ω_i|^1/2; (ii)he_i, e_ji= 0 ifi6=j; (iii) for every i, he_i, e_ii=ω_i; and (iv) everyu∈E_ω can be written uniquely as

u=

∞

X

i=0

u_ie_i, where u_i∈K, and lim

i→∞

|u_i|.|ω_i|^1/2

= 0.

The base (e_i)_i∈N is called the canonicalorthogonal base of E_ω.

In the literature, the spaceE_ω endowed with its norm and inner product given above, is called ap-adic or non-Archimedean Hilbert space. However, one should point out that the norm onE_ω does not stem from the inner product. In addition to that the spaceE_ω contains isotropic vectors, that is,hu, ui= 0 while 06=u∈ E_ω.

A bilinear formϕ : D(ϕ)×D(ϕ) 7→K with domain D(ϕ) is said to be rep- resentable (Definition 3.7) whenever there exists a (possibly unbounded) linear operatorA:D(A)7→E_ω (D(A) being the domain ofA) such that

ϕ(u, v) =hAu, vi, ∀u∈D(A), v∈D(ϕ).

An unbounded bilinear formϕ:D(ϕ)×D(ϕ)7→Kwhose domainD(ϕ) con- tains all elements of the canonical base (e_i)_i∈N is called stable. The subclass of all these stable unbounded bilinear forms is denoted Σ_S(E_ω×E_ω). Similarly, the subclass of all bilinear forms whose domains do not contain the above-mentioned canonical base is calledunstable and denoted Σ_U(E_ω×E_ω).

In Diagana [3], it was shown that ifϕis a non-degenerate, symmetric bilinear form satisfying

(1.1) lim

i→∞

|ϕ(e_i, e_j)|

ke_ik

= 0, ∀j∈N,

thenϕis uniquely representable. Moreover, ifAdenotes the (possibly unbounded) linear operator associated withϕ, then its adjointA^∗ does exist withA=A^∗.

In this paper we are interested in studying representation theorems for bounded and stable unbounded bilinear forms not necessarily symmetric. Namely, it is shown that a non-degenerate (stable) bilinear formϕonE_ω×E_ω is representable whenever

(1.2) lim

i→∞

|ϕ(e_i, e_j)|

keik

= lim

i→∞

|ϕ(e_j, e_i)|

keik

= 0, ∀j ∈N.

Moreover, ifAdenotes the linear operator onE_ω associated with the formϕ, then the adjointA^∗ ofA does exist.

Beside of the above-mentioned representation results for bilinear forms, we also establish a non-Archimedean version of the Riesz’s representation theorem for a subclass of linear functionals onE_ω. Namely, it is shown that ifF :E_ω7→Kis a linear functional such that

i→∞lim

|F(e_i)|

ke_ik = 0,

then there exists a unique vector u₀ ∈ E_ω such that F(u) = hu, u₀i for each u∈E_ω. Furthermore,|kFk|=ku₀k, where |k · k|is the natural norm onE^∗_ω, the (topological) dual ofE_ω.

Representing (un)bounded bilinear forms by linear operators in the classical setting is a topic that arises in several fields such as quantum mechanics through the study of form sums associated with the Hamiltonian, mathematical physics, symplectic geometry, and the study of weak solutions to some linear partial differ- ential equations, see, e.g., [2], [7], [11], [12]. In the non-Archimedean realm, one may expect some related applications in: (i) the study of weak solutions to some p-adic partial differential equations; and (ii) the study of a non-Archimedean ver- sion of the square root problem of Kato, which is of a great interest to the second named author.

To deal with the above-mentioned issues we shall make extensive use of the formalism of unbounded linear operators on non-Archimedean Hilbert spacesE_ω [4], [5], [8] and that of (un)bounded bilinear forms onE_ω×E_ω, recently introduced in [6] while studying non-Archimedean counterparts of the convergence in the sense of quadratic forms of bilinear forms defined on a Hilbert space.

3. Preliminary results

LetKbe a complete non-Archimedean valued field and let ω = (ω_i)_i∈N be a sequence of nonzero elements inK. Throughout the rest of the paper,E_ωdenotes the non-Archimedean Hilbert space associated with the sequence ω = (ω_i)_i∈N, and (e_i)_i∈Nstands for the canonical orthogonal base associated withE_ω. Define e^′_j∈E^∗_ω for eachj∈Nby

x=X

i∈N

x_ie_i∈E_ω, e^′_j(x) =x_j.

Definition 2.1([4], [5], [8]). A stable unbounded linear operatorAfromE_ωinto E_ω is a pair (D(A), A) consisting of a subspace D(A)⊂ E_ω (called the domain of A) and a (possibly not continuous) linear transformationA : D(A)⊂E_ω 7→

E_ω. Namely, the domain D(A) contains the basis (e_i)_i∈N and consists of all u= (u_i)_i∈N∈E_ω suchAu=P

i∈Nu_iAe_i converges inE_ω, that is,







D(A) :=

u= (u_i)_i∈N∈E_ω: lim

i→∞|u_i| kAe_ik= 0 , Au= X

i,j∈N

a_ije^′_j(u)e_i for each u∈D(A).

Definition 2.2 ([4], [5], [8]). A stable linear operator







D(A) :=

u= (u_i)_i∈N∈E_ω: lim

i→∞|u_i| kAe_ik= 0 , Au= X

i,j∈N

a_ije^′_j(u)e_i for each u∈D(A),

is said to have an adjointA^∗ if and only if

(2.1) lim

j→∞

|a_ij|

|ω_j|^1/2

!

= 0, ∀i∈N.

In this case the adjointA^∗ ofAis uniquely expressed by







D(A^∗) :=

v= (v_i)_i∈N∈E_ω: lim

i→∞|v_i| kA^∗e_ik= 0 , A^∗u= X

i,j∈N

a^∗_ij e^′_j(u)e_i for each u∈D(A^∗),

wherea^∗_ij = ^ωj_ω^aji

i .

Remark 2.3. (i) In contrast with the classical context, in the non-Archimedean setting, there are linear operators, which do not have adjoint operators.

(ii) In the classical setting, if A is a closable unbounded linear operator on a Hilbert space, then A^∗∗ = A, where A is the closure of A. However, in the non-Archimedean setting, if the adjointA^∗ of a stable unbounded linear operator Aexists, then A^∗∗=A.

2.1 Continuous linear functionals on E_ω

Definition 2.4. A linear functionalF :E_ω 7→Kis said to be continuous if there existsK≥0 such that

|F(u)| ≤K .kuk for each u∈E_ω.

The smallest constantK such that the previous inequality holds is called the norm of the continuous linear functionalF and is defined by

|kFk|= sup

u6=0

|F(u)|

kuk

.

Let us remind that the space of all continuous linear functionals on E_ω is denoted byE^∗_ω and called the (topological) dual ofE_ω. The space (E^∗_ω,|k · k|) is a Banach space overK.

Proposition 2.5. LetF∈E^∗_ω. Then its norm|kFk|can be explicitly expressed as

|kFk|= sup

i∈N

|F(e_i)|

ke_ik

.

The next theorem constitutes a non-Archimedean version of the well-known Riesz representation theorem [12].

Theorem 2.6. LetF :E_ω7→Kbe a linear functional such that

(2.2) lim

i→∞

|F(e_i)|

ke_ik

= 0.

Then there exists a uniqueu₀∈E_ω such that

F(u) =hu, u₀i, for all u∈E_ω. Moreover,|kFk|=ku₀k.

Proof: Obviously, (2.2) yieldsF is continuous, as|kFk|= sup_i∈N|F(ei)|

keik <∞.

Letu=P

i∈Nu_ie_i ∈E_ω. Now,F(u) =P

i∈Nu_iF(e_i) is well-defined. Indeed, sinceu∈E_ω, that is, lim_i→∞|u_i|ke_ik= 0, we have

i→∞lim |u_iF(e_i)| ≤ kuk. lim

i→∞

|F(e_i)|

ke_ik

= 0, by using assumption (2.2).

Now, setu₀ =P

i∈N(^F_ω^(ei)

i )e_i. Again using assumption (2.2), one can easily see thatu₀∈E_ω. Moreover,F(u) =hu, u₀ifor eachu∈E_ω.

Suppose that there exists another v₀ ∈ E_ω such that F(u) =hu, v₀i for each u ∈ E_ω. Then, hu₀ −v₀, ui = 0 for each u ∈ E_ω, that is, u₀−v₀⊥ E_ω. In particular,hu₀−v₀, e_ii= 0 for eachi∈N, that is, all coordinates ofu₀−v₀ in the canonical base (e_i)_i∈NofE_ω are zero, and hence u₀=v₀.

Now

ku₀k:= sup

i∈N

F(e_i) ω_i e_i

= sup

i∈N

|F(e_i)|

ke_ik =|kFk|.

3. Bilinear forms on E_ω×E_ω

Definition 3.1. A mapping ϕ : E_ω ×E_ω 7→ K is said to be a bilinear form wheneveru7→ϕ(u, v) is linear for eachv ∈ E_ω and v 7→ϕ(u, v) linear for each u∈E_ω.

Note that ifϕ:E_ω×E_ω 7→Kis a bilinear form overE_ω×E_ω, then the sum

(3.1) ϕ(u, v) =

∞

X

i,j=0

Ωiju_iv_j

may or may not be convergent. However if bothu= (u_i)_i∈Nandv= (v_i)_i∈Nare taken inE_ω with

i,j→∞lim

|u_i|.|Ω_ij|^1/2

= 0 and lim

i,j→∞

|v_j|.|Ω_ij|^1/2

= 0, where Ω_ij =ϕ(e_i, e_j) for alli, j∈N, then the sum in (3.1) converges.

3.1 Bounded bilinear forms

Definition 3.2. A non-Archimedean bilinear formϕ: E_ω×E_ω 7→Kis said to be bounded if there existsM ≥0 such that

(3.2) |ϕ(u, v)| ≤M .kuk.kvk for all u, v∈E_ω.

The smallestM such that (3.2) holds is called the norm of the bilinear formϕ and is defined by

kϕk= sup

u,v6=0

|ϕ(u, v)|

kuk.kvk

.

Proposition 3.3. Letϕ:E_ω×E_ω 7→Kbe a bounded bilinear form. Then its normkϕk can be explicitly expressed as

kϕk= sup

i,j∈N

|ϕ(e_i, e_j)|

ke_ik.ke_jk

.

Proof: The inequality, kϕk ≥ sup_i,j∈N(_ke^|ϕ(ei,ej)|

ik.kejk), is a straightforward conse- quence of the definition of the normkϕkofϕ.

Now supposeu, v6= 0. In view of the above, one has

|ϕ(u, v)|=

∞

X

i,j=0

ϕ(e_i, e_j)u_iv_j

≤ sup

i,j∈N

|ϕ(e_i, e_j)|.|u_i|.|v_j|

= sup

i,j∈N

|ϕ(e_i, e_j)|(|u_i|.ke_ik) (|v_j|.ke_jk) ke_ik.ke_jk

≤ kuk.kvk. sup

i,j∈N

|ϕ(e_i, e_j)|

keik.kejk

and hence

kϕk ≤ sup

i,j∈N

|ϕ(e_i, e_j)|

ke_ik.ke_jk

.

One completes the proof by combining the first and the last inequalities.

Definition 3.4. A bounded bilinear form ϕ: E_ω×E_ω 7→ Kis said to be rep- resentable whether there exists a bounded linear operator A : E_ω 7→ E_ω such that

ϕ(u, v) =hAu, vi, ∀u, v∈E_ω.

Theorem 3.5. Letϕ:E_ω×E_ω7→Kbe a non-degenerate bounded bilinear form on E_ω×E_ω. Then ϕ is representable whenever (1.2) holds. In this case, if A denotes the linear operator associated withϕ, then the adjointA^∗ of Aexists.

Proof: Define the linear operatorAonE_ω associated withϕas follows:

Au:= X

i,j∈N

ϕ(e_j, e_i) ω_i

e^′_j(u)e_i

for eachu∈E_ω.

We first check that the linear operatorA given above is well-defined on E_ω. For that, it suffices to see that, for allj ∈N,

i→∞lim

ϕ(e_j, e_i) ω_i

ke_ik= lim

i→∞

|ϕ(e_i, e_j)|

ke_ik = 0,

by using assumption (1.2). Furthermore, it is routine to see thatϕ(u, v) =hAu, vi for allu, v∈E_ω. Of course, the linear operatorA given above is unique sinceφ is non-degenerate.

Now

kAk:= sup

i,j∈N





ϕ(ej,ei) ωi

ke_ik e_j



= sup

i,j∈N

|ϕ(e_j, e_i)|

ke_jk.ke_ik

=kϕk,

and henceA is bounded.

It remains to show thatA^∗, the adjoint ofAexists. Indeed,

j→∞lim





ϕ(ej,ei) ωi

ke_jk



= 1

|ω_i|. lim

j→∞

|ϕ(ej, e_i)|

ke_jk

= 0, ∀i∈N,

by using assumption (1.2), and hence the adjointA^∗ ofAexists.

Example 3.6. Let (K,| · |) = (Q_p,| · |) equipped with the p-adic absolute value and letω_i=p⁻ⁱ for eachi∈N. LetN₀∈NwithN₀≥1 (fixed) and set

π^N_ij⁰ = 1 + 1 ω_j + 1

ω_i²ω_j² +· · ·+ 1 ω_i^N0ω_j^N0 for alli, j∈N.

Now,∀j∈N, lim_i→∞ ^|π

N0 ij |

keik = lim_i→∞ ^|π

N0 ji |

keik = 0, since|π_ij^N0|=|π^N_ji⁰|= 1 and ke_ik=p^i/2for alli∈N. For allu= (u_i)_i∈N, v= (v_i)_i∈N∈E_ω, define the bilinear form as follows:

ϕ(u, v) =

∞

X

i,j=0

π^N_ij⁰ u_iv_j.

Obviously,ϕis well-defined since,∀j∈N,

i→∞lim

|u_i|.|π^N_ij⁰|^1/2

≤ kuk. lim

i→∞

1 ke_ik = 0.

Moreover ϕ is non-degenerate and its norm kϕk = 1. Therefore, the only bounded linear operator onE_ω associated withϕis the one defined by

Au= X

i,j∈N

"

π_ji^N0 ω_i

# e^′_j(u)e_i

for eachu∈E_ω withkAk= sup_i,j∈N( ^|π

N0 ij |

keik.kejk) = 1.

It is also clear thatA^∗, the adjoint ofAexists.

3.2 Stable unbounded bilinear forms

In this subsection we present with a representation theorem for some un- bounded bilinear forms. More precisely, we consider those unbounded bilinear forms whose domains contain all elements of the canonical base (ei)_i∈NofE_ω, as such a base plays a key role in the present setting. The subclass of all those types of unbounded bilinear forms will be called stable and denoted by Σ_S(E_ω×E_ω).

Similarly, the subclass of all unbounded bilinear forms whose domains do not contain elements of the above-mentioned canonical base will be called unstable and denoted by Σ_U(E_ω ×E_ω). Note that a representation theorem similar to Theorem 3.9 for elements of Σ_U(E_ω×E_ω) will be left as an open question.

Definition 3.7. A mapping ϕ : D(ϕ)×D(ϕ) ⊂ E_ω ×E_ω 7→ K is said to be a stable unbounded bilinear form if u 7→ ϕ(u, v) is linear for each v ∈ D(ϕ),

v 7→ϕ(u, v) is linear for each u∈D(ϕ), whereD(ϕ) is a vector subspace of E_ω that contains the base (e_i)_i∈N, and



















D(ϕ) :=

u= (u_i)_i∈N∈E_ω: lim

i,j→∞

|u_i| |Ω_ij|^1/2

= lim

i,j→∞

|u_i| |Ω_ji|^1/2

= 0 , ϕ(u, v) =

∞

X

i,j=0

Ω_ij u_iv_j, for all u, v∈D(ϕ),

where Ω_ij =ϕ(e_i, e_j).

The spaceD(ϕ) defined above is called thedomain of the bilinear formϕ.

Definition 3.8. A bilinear formϕ:D(ϕ)×D(ϕ)7→K(D(ϕ) being its domain) is said to be representable whenever there exists a (possibly unbounded) linear operatorA:D(A)7→E_ω (D(A) being the domain ofA) such that

ϕ(u, v) =hAu, vi, ∀u∈D(A), v∈D(ϕ).

Theorem 3.9. Letϕ:D(ϕ)×D(ϕ)7→Kbe a non-degenerate stable unbounded bilinear form. Thenϕis representable whenever assumption(1.2)holds. In this case, if A denotes the linear operator associated with ϕ, then the adjointA^∗ of Aexists.

Proof: For allu= (u_i)_i∈N,v= (v_j)_j∈N∈D(ϕ), writeϕ(u, v) =P_∞

i,j=0Ω_ij u_iv_j and define the linear operatorAonE_ω associated to it as follows:











D(A) :=

u= (u_i)_i∈N∈E_ω : lim

i→∞|u_i| kAe_ik= 0 , Au= X

i,j∈N

ϕ(e_j, e_i) ω_i

e^′_j(u)e_i for each u= (u_i)_i∈N∈D(A).

Obviously,Ais well-defined. Indeed, for allj∈N,

i→∞lim

ϕ(e_j, e_i) ω_i

ke_ik= lim

i→∞

|ϕ(e_j, e_i)|

ke_ik = lim

i→∞

|ϕ(e_i, e_j)|

ke_ik = 0, by using assumption (1.2).

Now

Au=X

i∈N

1 ω_i

X

j∈N

u_jϕ(e_j, e_i)

e_i for each u= (u_i)_i∈N∈D(A),

and hencehAe_i, e_ji=ϕ(e_j, e_i) for alli, j∈N.

Moreover, D(A) ⊂ D(ϕ). Indeed, if u = (u_i)_i∈N ∈ D(A), then using the Cauchy-Schwartz inequality it follows that,∀i, j∈N,

|u_i|.|u_j|.|ϕ(e_i, e_j)|=|u_i|.|u_j|.|hAe_j, e_ii|

≤(|u_j|.kAe_jk).(ke_ik.|u_i|), and hence

i,j→∞lim |u_i|.|ϕ(e_i, e_j)|^1/2 2

≤ lim

i,j→∞(|u_j|.kAe_jk).(ke_ik.|u_i|)

= 0, that is,u∈D(ϕ).

Note thatu_iv_kϕ(e_i, e_k)→0 asi, k→ ∞, by using the fact that (u∈D(A)⊂ D(ϕ) andv∈D(ϕ)):

|u_iv_kϕ(e_i, e_k)|=

|u_i||ϕ(e_i, e_k)|^1/2 .

|ϕ(e_i, e_k)|^1/2|v_k|

→0, as i, k→ ∞, and hence

X

k∈N

X

i∈N

u_iv_kϕ(e_i, e_k) =X

i∈N

X

k∈N

u_iv_kϕ(e_i, e_k).

Consequently, the following successive equalities are justified:

hAu, vi=X

k∈N

ω_kv_k 1 ω_k

X

i∈N

u_iϕ(e_i, e_k)

=X

k∈N

v_k

X

i∈N

u_iϕ(e_i, e_k)

= X

i,k∈N

ϕ(e_i, e_k)u_iv_k

=ϕ(u, v)

for allu= (u_i)_i∈N∈D(A) andv= (v_i)_i∈N∈D(ϕ).

Furthermore, the uniqueness of A is guaranteed by the fact that ϕ is non- degenerate. It remains to show thatA^∗, the adjoint ofAexists; however, this can

be done as in the bounded case.

Example 3.10. This example is a generalization of Example 3.6. Consider the bilinear form defined by

ϕ(u, v) = X

i,j∈N

π_ij. u_iv_j, ∀u= (u_i)_i∈N, v= (v_i)_i∈N∈D(ϕ),

where (π_ij)_i,j∈N ⊂ K is an arbitrary sequence, and the domain D(ϕ) of ϕ is defined by

D(ϕ) =

u= (u_i)_i∈N∈E_ω : lim

i,j→∞

|u_i|.|π_ij|^1/2

= lim

i,j→∞

|u_i|.|π_ji|^1/2

= 0

. Note that ϕ(e_i, e_j) = π_ij for all i, j ∈ N and hence an equivalent of assump- tion (1.2) is:

(3.3) lim

i→∞

|π_ij| ke_ik = lim

i→∞

|π_ji| ke_ik = 0.

Upon making assumption (3.3), the unique (possibly unbounded) linear ope- rator associated withϕis given by

Au= X

i,j∈N

π_ji

ω_i e^′_j(u)e_i, ∀u= (u_i)_i∈N∈D(A), whereD(A) ={u= (u_i)_i∈N∈E_ω: lim_i→∞(kAe_ik.|u_i|) = 0}.

In addition to the above, the adjointA^∗ofAdoes exist under assumption (3.3).

Acknowledgment. The authors express their thanks to the referee for his/her valuable comments and suggestions on the paper.

References

[1] Basu S., Diagana T., Ramaroson F., Ap-adic version of Hilbert-Schmidt operators and applications, J. Anal. Appl.2(2004), no. 3, 173–188.

[2] de Bivar-Weinholtz A., Lapidus M.L.,Product formula for resolvents of normal operator and the modified Feynman integral, Proc. Amer. Math. Soc.110(1990), no. 2, 449–460.

[3] Diagana T.,Representation of bilinear forms in non-Archimedean Hilbert space by linear operators, Comment. Math. Univ. Carolin.47(2006), no. 4, 695–705.

[4] Diagana T.,Towards a theory of some unbounded linear operators onp-adic Hilbert spaces and applications, Ann. Math. Blaise Pascal12(2005), no. 1, 205–222.

[5] Diagana T.,Erratum to: “Towards a theory of some unbounded linear operators onp-adic Hilbert spaces and applications”, Ann. Math. Blaise Pascal13(2006), 105–106.

[6] Diagana T.,Bilinear forms on non-Archimedean Hilbert spaces, preprint, 2005.

[7] Diagana T.,Fractional powers of the algebraic sum of normal operators, Proc. Amer. Math.

Soc.134(2006), no. 6, 1777–1782.

[8] Diagana T., An Introduction to Classical and p-adic Theory of Linear Operators and Applications, Nova Science Publishers, New York, 2006.

[9] Diarra B.,An operator on some ultrametric Hilbert spaces, J. Analysis6(1998), 55–74.

[10] Diarra B.,Geometry of thep-adic Hilbert spaces, preprint, 1999.

[11] Johnson G.W., Lapidus M.L.,The Feynman Integral and Feynman Operational Calculus, Oxford Univ. Press, Oxford, 2000.

[12] Kato T.,Perturbation Theory for Linear Operators, Springer, New York, 1966.

[13] Ochsenius H., Schikhof W.H.,Banach Spaces Over Fields with an Infinite Rank Valuation, p-adic Functional Analysis, (Poznan, 1998), Marcel Dekker, New York, 1999, pp. 233–293.

[14] van Rooij A.C.M.,Non-Archimedean Functional Analysis, Marcel Dekker, New York, 1978.

Department of Mathematics, Howard University, 2441 6th Street N.W., Washington, D.C. 20059, USA

E-mail: [email protected]

Department of Mathematics, Howard University, 2441 6th Street N.W., Washington, D.C. 20059, USA

E-mail: [email protected]

(Received August 11, 2006,revised March 6, 2007)