On metrizability of compactoid sets in non-archimedean locally convex spaces

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ON METRIZABILITY OF COMPACTOID SETS

IN NON-ARCHIMEDEAN LOCALLY CONVEX

SPACES

JERZY KA¸ KOL and WIES LAW ´SLIWA

UNIVERSITY OF POZNA ´N, POLAND

10th International Conference on P-Adic and Non-Archimedean Analysis, Michigan State University, East Lansing, Michigan,

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Introduction

In our presentation all locally convex spaces (lcs) are non-archimedean and Hausdorff.

In 2003 N. De Grande-De Kimpe, J. K¸akol and C.Perez-Garcia using t-frames and some machinery concerning tensor products proved that compactoid sets in (LM)-spaces (i.e. the inductive limits of a sequence of metrizable locally convex spaces) are metrizable ([GKP, Theorem 3.1]).

In this paper we show a similar result for locally convex spaces with a L-base, i.e. with a decreasing base (Uα)α∈NN of

neighbourhoods of zero. This extends the first mentioned result since every (LM)-space has a L-base.

We also prove that compactoid sets in (DF )-spaces are metrizable.

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Introduction

We also prove that compactoid sets in (DF )-spaces are metrizable.

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Introduction

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We start with the following useful fact ([KS, Lemma 1]). Lemma 1

Let A be an absolutely convex compactoid set in a

polar lcs E . Then for every c > 0 and every f ∈ [A]

0

there is a g ∈ E

0

with |f (a) − g (a)| < c for a ∈ A.

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Sketch of Proof

Put H = [A]. Let c > 0 and let f ∈ H0.

Then for some continuous polar seminorm p on E we have |f (x)| ≤ p(x) for every x ∈ H. Put

U = {x ∈ E : p(x ) < c/(c + 1)}.

By [Kat, Theorem 4.1] there exists a finite dimensional subspace D of H with A ⊂ U + D.

By [Per1, Corollary 2.2] there is a g ∈ E0 with g |D = f |D such that

|g (x)| ≤ (c + 1)p(x) for all x ∈ E .

Let a ∈ A. Then a is of the form u + d , where u ∈ U and d ∈ D. Hence u ∈ H and

|f (a) − g (a)| = |f (u) − g (u)| ≤ (c + 1)p(u) < c. This completes the proof.

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Sketch of Proof

U = {x ∈ E : p(x ) < c/(c + 1)}. By [Kat, Theorem 4.1] there exists a finite dimensional subspace D of H with A ⊂ U + D.

|g (x)| ≤ (c + 1)p(x) for all x ∈ E .

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Sketch of Proof

|g (x)| ≤ (c + 1)p(x) for all x ∈ E .

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Sketch of Proof

|g (x)| ≤ (c + 1)p(x) for all x ∈ E .

Let a ∈ A. Then a is of the form u + d , where u ∈ U and

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Sketch of Proof

|g (x)| ≤ (c + 1)p(x) for all x ∈ E .

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Denote by E_c0 the dual of E with the topology c(E0, E ) of uniform convergence on compactoid subsets of E .

Using Lemma 1 we shall prove the following ([KS, Theorem 2])

Theorem 2

For a polar lcs E the following conditions are

equivalent.

(a) Every compactoid set in E is metrizable.

(b) E

_c0

is of countable type.

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Denote by E_c0 the dual of E with the topology c(E0, E ) of uniform convergence on compactoid subsets of E .

Using Lemma 1 we shall prove the following ([KS, Theorem 2])

Theorem 2

For a polar lcs E the following conditions are

equivalent.

(a) Every compactoid set in E is metrizable.

(b) E

_c0

is of countable type.

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Sketch of Proof

(a) ⇒ (b).

Let G = E_c0 and let p ∈ P(G ). Then for some metrizable absolutely convex compactoid set A in E we have p ≤ pA,

where pA : G → [0, ∞), pA(f ) = supx ∈A|f (x)|.

Let α ∈ K with |α| > 1. By [Sch1, Proposition 8.2] there exists a sequence (xn) ⊂ αA with xn → 0 in E such that its

closed absolutely convex hull X includes A. Let H = ker pA. The operator

T : G /H → c0, T (f + H) = (f (xn)),

is well defined and linear. For f ∈ G we have

max

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Sketch of Proof

(a) ⇒ (b).

closed absolutely convex hull X includes A.

Let H = ker pA. The operator

T : G /H → c0, T (f + H) = (f (xn)),

max

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Sketch of Proof

(a) ⇒ (b).

T : G /H → c0, T (f + H) = (f (xn)),

is well defined and linear.

For f ∈ G we have max

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Sketch of Proof

(a) ⇒ (b).

T : G /H → c0, T (f + H) = (f (xn)),

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Sketch of Proof

Hence we get

kT (f + H)k∞≤ |α|pA(f + H) ≤ |α|kT (f + H)k∞.

Thus the normed space M = (G /H, pA) is isomorphic to a

subspace of c0, so it is of countable type.

Let S = {fn+ H : n ∈ N} be a linearly dense countable subset

of M.

Then the set W = {fn+ ker p : n ∈ N} is linearly dense in the

normed space Gp= (G / ker p, p).

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Sketch of Proof

Hence we get

of M.

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Sketch of Proof

Hence we get

of M.

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Sketch of Proof

Hence we get

of M.

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Sketch of Proof

Hence we get

of M.

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Sketch of Proof

(b) ⇒ (a).

Let A be an absolutely convex compactoid set in E . Let G , pA, H and S = {fn+ H : n ∈ N} be as above.

Let F = [A]. F is of countable type ([Sch2, Proposition 4.3]). Using [Kat, Theorem 4.1] we infer that A is compactoid in F . Clearly the functional

k · kA : F0 → [0, ∞), g → sup x ∈A

|g (x)| is a norm on F0.

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Sketch of Proof

(b) ⇒ (a).

Let A be an absolutely convex compactoid set in E .

Let G , pA, H and S = {fn+ H : n ∈ N} be as above.

k · kA : F0 → [0, ∞), g → sup x ∈A

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Sketch of Proof

(b) ⇒ (a).

k · kA : F0 → [0, ∞), g → sup x ∈A

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Sketch of Proof

(b) ⇒ (a).

Let F = [A]. F is of countable type ([Sch2, Proposition 4.3]).

Using [Kat, Theorem 4.1] we infer that A is compactoid in F . Clearly the functional

k · kA : F0 → [0, ∞), g → sup x ∈A

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Sketch of Proof

(b) ⇒ (a).

Let F = [A]. F is of countable type ([Sch2, Proposition 4.3]). Using [Kat, Theorem 4.1] we infer that A is compactoid in F .

Clearly the functional

k · kA : F0 → [0, ∞), g → sup x ∈A

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Sketch of Proof

(b) ⇒ (a).

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Sketch of Proof

Applying Lemma 1 one can show that the set L = {fn|F : n ∈ N} is linearly dense in (F0, k · kA).

Indeed, let f ∈ F0 and c > 0.

By Lemma 1 there exists g ∈ E0 with kf − g |F kA ≤ c.

Moreover, there exists h ∈ [{fn : n ∈ N}] with pA(g − h) ≤ c.

Hence kf − h|F kA ≤ c; clearly h|F ∈ [L].

Thus (F0, k · kA) is of countable type.

Using [GKP, Lemma 1.1] we infer that A is metrizable in E . It follows that all compactoid sets in E are metrizable.

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Sketch of Proof

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Sketch of Proof

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Sketch of Proof

Using [GKP, Lemma 1.1] we infer that A is metrizable in E .

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Sketch of Proof

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Let E be a polar lcs. Since Eσ = (E , σ(E , E0)) is of finite type,

it has no subspace isomorphic to c0; so every bounded set in

Eσ is compactoid ([GKPS3, Corollary 6.7]).

It follows that (Eσ)0c = Eb0. Thus, using Theorem 2 for Eσ,

we get the following known result ([Sch1, Theorem 8.3]): Corollary 3

The strong dual E

_b0

of a polar lcs E is of countable

type if and only if every bounded set in E is

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we get the following known result ([Sch1, Theorem 8.3]):

Corollary 3

The strong dual E

_b0

of a polar lcs E is of countable

type if and only if every bounded set in E is

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we get the following known result ([Sch1, Theorem 8.3]): Corollary 3

The strong dual E

_b0

of a polar lcs E is of countable

type if and only if every bounded set in E is

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By Theorem 2 and its proof we get as well ([KS, Corollary 3]) Corollary 4

For a lcs E the following conditions are equivalent.

(a) Every compactoid set in E is metrizable.

(b) For every subspace F of countable type in E the

space F

_c0

is of countable type.

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Definition of a resolution

For α = (αn), β = (βn) ∈ NN we write α ≤ β if αn≤ βn for

all n ∈ N.

A family (Aα)α∈NN of sets is increasing if Aα ⊂ Aβ for all

α, β ∈ NN _{with α ≤ β.}

Similarly we define a decreasing family (Aα)α∈NN of sets.

By a resolution of a set A we mean an increasing family (Aα)α∈NN of subsets of A with S{Aα : α ∈ NN} = A.

A resolution (Aα)α∈NN of a lcs E is compactoid [bounded] if

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Definition of a resolution

all n ∈ N.

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Definition of a resolution

all n ∈ N.

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Definition of a resolution

all n ∈ N.

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We will use the following known fact ([MS, Lemma 2.1]). Lemma A

Let (A

α

)

_α∈NN

be a resolution of an uncountable set

A. Then for some β ∈ N

N

_{the set A}

β

is infinite.

Using the concept of a t-frame in a normed space we show the following ([KS, Proposition 5])

Proposition 5

Any lcs E with a compactoid resolution (A

α

)

_α∈NN

is

of countable type.

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We will use the following known fact ([MS, Lemma 2.1]). Lemma A

Let (A

α

)

_α∈NN

be a resolution of an uncountable set

A. Then for some β ∈ N

N

_{the set A}

β

is infinite.

Using the concept of a t-frame in a normed space we show the following ([KS, Proposition 5])

Proposition 5

Any lcs E with a compactoid resolution (A

α

)

_α∈NN

is

of countable type.

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Sketch of Proof

Let p ∈ P(E ), Ep = E / ker p and let πp : E → Ep be the

quotient map.

Let Cα = πp(Aα) for α ∈ NN. Then (Cα)α∈NN is a compactoid

resolution of the normed space Ep = (Ep, p).

Fix t ∈ (0, 1). Suppose that Ep contains an uncountable

t-frame X ; without loss of generality we may assume that p(x ) ≥ 1 for all x ∈ X .

By Lemma A, for some β ∈ NN _{the set B}

β = Cβ∩ X is

infinite. The infinite compactoid t-frame Bβ in the normed

space Ep is distant from 0.

It is in contradiction with [GPS, Proposition 2.2].

Thus every t-frame in the normed space Ep is countable.

Applying [GPS, Theorem 2.5] we deduce that the space Ep is

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Sketch of Proof

quotient map.

β = Cβ∩ X is

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Sketch of Proof

quotient map.

β = Cβ∩ X is

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Sketch of Proof

quotient map.

β = Cβ∩ X is

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Sketch of Proof

quotient map.

β = Cβ∩ X is

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Using Proposition 5 we get ([KS, Corollary 6]) Corollary 6

If every bounded subset of a metrizable lcs E is

compactoid, then E is of countable type.

PROOF. Let (Un) be a base of absolutely convex

neighbourhoods of zero in E .

Let β ∈ K with |β| > 1. For α = (αn) ∈ NN we put

Aα = ∞

\

n=1

βαnUn.

Clearly, (Aα)α∈NN is a bounded resolution of E ; so (Aα)α∈NN is

a compactoid resolution of E .

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If every bounded subset of a metrizable lcs E is

compactoid, then E is of countable type.

Aα = ∞

\

n=1

βαnUn.

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If every bounded subset of a metrizable lcs E is

compactoid, then E is of countable type.

Aα = ∞

\

n=1

βαnUn.

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If every bounded subset of a metrizable lcs E is

compactoid, then E is of countable type.

Aα = ∞

\

n=1

βαnUn.

Clearly, (A ) is a bounded resolution of E ; so (A ) is

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If every bounded subset of a metrizable lcs E is

compactoid, then E is of countable type.

Aα = ∞

\

n=1

βαnUn.

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Last result yields the following known fact ([GPS, Theorem 3.1]).

Corollary 7

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A resolution (Aα)α∈NN of a lcs E is countably compactoid

if every countable subset of Aα, α ∈ NN, is compactoid.

Following the proof of Proposition 5 we get the following. Remark 8

Any lcs E with a countably compactoid resolution is

of countable type.

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A resolution (Aα)α∈NN of a lcs E is countably compactoid

if every countable subset of Aα, α ∈ NN, is compactoid.

Following the proof of Proposition 5 we get the following. Remark 8

Any lcs E with a countably compactoid resolution is

of countable type.

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We will need the following fact which follows from [Sch1, Lemma 10.6] and [Sch2, Proposition 4.5].

Lemma 9

For a lcs E every equicontinuous set A in E

0

is

compactoid in E

_c0

.

PROOF. There is a neighbourhood U of zero in E such that A ⊂ U◦. The set U◦ is compactoid in E_σ0 ([Per2, Theorem 4.2]).

Applying [Sch1, Lemma 10.6] one gets that the topologies σ(E0, E ) and c(E0, E ) coincide on U◦.

Finally using [Sch2, Proposition 4.5] we deduce that the set U◦ is compactoid in E_c0_.

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We will need the following fact which follows from [Sch1, Lemma 10.6] and [Sch2, Proposition 4.5].

Lemma 9

For a lcs E every equicontinuous set A in E

0

is

compactoid in E

_c0

.

PROOF. There is a neighbourhood U of zero in E such that A ⊂ U◦. The set U◦ is compactoid in E_σ0 ([Per2, Theorem 4.2]).

Applying [Sch1, Lemma 10.6] one gets that the topologies σ(E0, E ) and c(E0, E ) coincide on U◦.

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Making use of Lemma 9, Proposition 5 and Corollary 4 we get ([KS, Theorem 10])

Theorem 10

Let E be a lcs with a L-base i.e. with a decreasing

base (U

α

)

_α∈NN

of neighbourhoods of zero.

Then every compactoid set in E is metrizable.

PROOF. Let F be a subspace of countable type in E . Let Vα = Uα∩ F for α ∈ NN. Clearly, (Vα)α∈NN is a L-base in F .

Then (V_α◦)_α∈NN is a compactoid resolution of F_c0.

By Proposition 5, the space F_c0 is of countable type.

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Theorem 10

Let E be a lcs with a L-base i.e. with a decreasing

base (U

α

)

_α∈NN

of neighbourhoods of zero.

Then every compactoid set in E is metrizable.

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Theorem 10

Let E be a lcs with a L-base i.e. with a decreasing

base (U

α

)

_α∈NN

of neighbourhoods of zero.

Then every compactoid set in E is metrizable.

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Theorem 10

Let E be a lcs with a L-base i.e. with a decreasing

base (U

α

)

_α∈NN

of neighbourhoods of zero.

Then every compactoid set in E is metrizable.

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Applying Theorem 10 we get the following well-known result ([GKP, Theorem 3.1]).

Corollary 11

Every compactoid set in a (LM)-space E is

metrizable.

PROOF. Let (En) be an inductive sequence of metrizable lcs

such that E = lim₋_→En.

Let (Un,k)∞k=1 be a decreasing base of absolutely convex

neighbourhoods of zero in En for n ∈ N. Put

Vα = ∞ X n=1 Un,αn(= ∞ [ i =1 i X n=1 Un,αn) for α = (αn) ∈ NN.

Using [GKPS1, Proposition 1.1.7(i)] one gets easily that (Vα)α∈NN is a L-base in E .

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Corollary 11

Every compactoid set in a (LM)-space E is

metrizable.

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Corollary 11

Every compactoid set in a (LM)-space E is

metrizable.

neighbourhoods of zero in En for n ∈ N.

Put Vα = ∞ X n=1 Un,αn(= ∞ [ i =1 i X n=1 Un,αn) for α = (αn) ∈ NN.

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Corollary 11

Every compactoid set in a (LM)-space E is

metrizable.

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Corollary 11

Every compactoid set in a (LM)-space E is

metrizable.

Using [GKPS1, Proposition 1.1.7(i)] one gets easily that (V ) is a L-base in E .

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Corollary 11

Every compactoid set in a (LM)-space E is

metrizable.

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We also get the following ([KS, Corollary 12]). Corollary 12

Let E be a metrizable lcs. Then every compactoid

set in the strong dual E

_b0

of E is metrizable.

PROOF. Let (Aα)α∈NN be the bounded resolution of E

constructed in the proof of Corollary 6 i.e. Aα =

∞

\

n=1

βαnUn

for α = (αn) ∈ NN, where (Un) is a base of absolutely convex

neighbourhoods of zero in E and β ∈ K with |β| > 1.

Clearly, for every bounded set A in E there exists α ∈ NN _such

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Let E be a metrizable lcs. Then every compactoid

set in the strong dual E

_b0

of E is metrizable.

∞

\

n=1

βαnUn

Clearly, for every bounded set A in E there exists α ∈ NN _such

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Let E be a metrizable lcs. Then every compactoid

set in the strong dual E

_b0

of E is metrizable.

∞

\

n=1

βαnUn

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Denote by L the family of all locally convex spaces E with a L-base (Uα)α∈NN. Clearly all metrizable lcs belong to L.

It is easy to check the following ([KS, Proposition 13]) Proposition 13

Let E ∈ L. Then every subspace F of E belongs to

L. If F is a closed subspace of E , then E /F ∈ L.

The completion of E belongs to L.

We show that the class L is stable by countable products, locally convex countable inductive and projective limits and locally convex countable direct sums ([KS, Propositions 14 and 15 and Corollaries 16 and 17]).

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Let E ∈ L. Then every subspace F of E belongs to

L. If F is a closed subspace of E , then E /F ∈ L.

The completion of E belongs to L.

We show that the class L is stable by countable products, locally convex countable inductive and projective limits and locally convex countable direct sums ([KS, Propositions 14 and 15 and Corollaries 16 and 17]).

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Let E ∈ L. Then every subspace F of E belongs to

L. If F is a closed subspace of E , then E /F ∈ L.

The completion of E belongs to L.

We show that the class L is stable by countable products, locally convex countable inductive and projective limits and

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Proposition 14

If (E

n

) ⊂ L, then E =

Q

∞

n=1

E

n

belongs to L.

Proposition 15

If (E

n

) ⊂ L is an inductive sequence, then lim

₋

_→

E

n

belongs to L.

Corollary 16

If (En) ⊂ L, then the locally convex direct sum E = ⊕∞n=1En

belongs to L.

Corollary 17

If (En) ⊂ L is a projective sequence of lcs, then the projective

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Proposition 14

If (E

n

) ⊂ L, then E =

Q

∞

n=1

E

n

belongs to L.

Proposition 15

If (E

n

) ⊂ L is an inductive sequence, then lim

₋

_→

E

n

belongs to L.

Corollary 16

belongs to L.

Corollary 17

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Proposition 14

If (E

n

) ⊂ L, then E =

Q

∞

n=1

E

n

belongs to L.

Proposition 15

If (E

n

) ⊂ L is an inductive sequence, then lim

₋

_→

E

n

belongs to L.

Corollary 16

belongs to L.

Corollary 17

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Proposition 14

If (E

n

) ⊂ L, then E =

Q

∞

n=1

E

n

belongs to L.

Proposition 15

If (E

n

) ⊂ L is an inductive sequence, then lim

₋

_→

E

n

belongs to L.

Corollary 16

belongs to L.

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Next we shall prove that compactoid sets in (DF )-spaces are metrizable.

Recall that a lcs E is a (DF )-space if it has a fundamental sequence (Bn) of bounded sets and for every sequence (Vn) of

absolutely convex neighbourhoods of zero in E such that the set V =T∞

n=1Vn is bornivorous, V is a neighbourhood of zero

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Next we shall prove that compactoid sets in (DF )-spaces are metrizable.

Recall that a lcs E is a (DF )-space if it has a fundamental sequence (Bn) of bounded sets and for every sequence (Vn) of

absolutely convex neighbourhoods of zero in E such that the set V =T∞

n=1Vn is bornivorous, V is a neighbourhood of zero

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One can show the following lemma ([KS, Lemma 18]) Lemma 18

If K is not spherically complete, L is the spherical

completion of K and E is a (DF )-space over K,

then the space H = L ⊗ E is a (DF )-space over L.

Using this lemma we prove the following ([KS, Theorem 19]) Theorem 19

Every compactoid set in a (DF )-space E is

metrizable.

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One can show the following lemma ([KS, Lemma 18]) Lemma 18

If K is not spherically complete, L is the spherical

completion of K and E is a (DF )-space over K,

then the space H = L ⊗ E is a (DF )-space over L.

Using this lemma we prove the following ([KS, Theorem 19]) Theorem 19

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Sketch of Proof

First we show that every countable bounded subset F = {fn: n ∈ N} of Eb0 is equicontinuous. Let Fn = {fi : 1 ≤ i ≤ n} for n ∈ N. Then ◦F = T∞ n=1 ◦_F n. Clearly ◦ Fn = n \ k=1 f_k−1(B_K)

is an absolutely convex neighbourhood of zero in E for n ∈ N. Moreover,◦F is a bornivorous set in E . Indeed, let B be a bounded set in E . Then for some α ∈ K we have F ⊂ αB◦, so

◦

F ⊃ α−1(◦(B◦)) ⊃ α−1B.

Thus◦F is a neighbourhood of zero in E , so F is equicontinuous.

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Sketch of Proof

Now we consider two cases:

(1) E is polar.

Let (Bn) be a fundamental sequence of bounded sets in E and

let β ∈ K with |β| > 1. Put Aα = ∞ \ k=1 βαkB_k◦ for α = (αk) ∈ NN.

Clearly (Aα)α∈NN is a bounded resolution of E_b0. By Lemma 9,

every countable subset of Aα, α ∈ NN, is compactoid in Ec0.

Thus E_c0 has a countably compactoid resolution. Using Remark 8 we deduce that E_c0 is of countable type.

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Sketch of Proof

Now we consider two cases: (1) E is polar.

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Sketch of Proof

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Sketch of Proof

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Sketch of Proof

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Sketch of Proof

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Sketch of Proof

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Sketch of Proof

(2) E is not polar.

Then K is not spherically complete. By Lemma 18, H = L ⊗ E is a polar (DF )-space over the field L. Let A be a compactoid set in E . Then C = 1 ⊗ A is a compactoid set in H.

By (1), the set C is a metrizable subset of H. But C is homeomorphic to A, so A is metrizable.

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Sketch of Proof

(2) E is not polar.

Then K is not spherically complete. By Lemma 18, H = L ⊗ E is a polar (DF )-space over the field L.

Let A be a compactoid set in E . Then C = 1 ⊗ A is a compactoid set in H.

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Sketch of Proof

(2) E is not polar.

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Sketch of Proof

(2) E is not polar.

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We say that a compactoid resolution (Sα)α∈NN of a lcs E is

strong if every compactoid set in E is contained in some Sα.

Finally, in connection to we shall prove ([KS, Theorem 21]) Theorem 20

Every metrizable lcs E of countable type has a

strong compactoid resolution (S

α

)

_α∈NN.

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We say that a compactoid resolution (Sα)α∈NN of a lcs E is

strong if every compactoid set in E is contained in some Sα.

Finally, in connection to Proposition 5 we shall prove ([KS, Theorem 21])

Theorem 20

Every metrizable lcs E of countable type has a

strong compactoid resolution (S

α

)

_α∈NN.

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Sketch of Proof

By [Gra2, Corollary 3.7], a subset A of c0 is compactoid if and

only if there exists y = (yn) ∈ c0 such that for every

x = (xn) ∈ A we have |xn| ≤ |yn| for all n ∈ N.

(A1). Assume that K is separable.

Then c0 is separable; so there is a homeomorphism

f : NN_{→ c}

0 ([Sli, Theorem 3]).

For α = (αn) ∈ NN we put

Aα = f ({β ∈ NN: β ≤ α});

Aα is compact, so it is compactoid in c0.

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Sketch of Proof

f : NN_{→ c}

Aα = f ({β ∈ NN: β ≤ α});

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Sketch of Proof

f : NN_{→ c}

Aα = f ({β ∈ NN: β ≤ α});

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Sketch of Proof

f : NN_{→ c}

Aα = f ({β ∈ NN: β ≤ α});

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Sketch of Proof

f : NN_{→ c}

Aα = f ({β ∈ NN: β ≤ α});

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Sketch of Proof

f : NN_{→ c}

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Sketch of Proof

Let α ∈ NN_{. Then there exists y}

α = (yα,n) ∈ c0 such that |xn| ≤ |yα,n| for all x = (xj) ∈ Aα, n ∈ N. Let sα,n = sup{|xn| : x = (xj) ∈ Aαfor n ∈ N}. Then sα,n ≤ |yα,n| for n ∈ N, so sα,n →n0. If α, β ∈ NN _{and α ≤ β, then s} α,n ≤ sβ,n for n ∈ N.

(A2). Now let the field K be arbitrary. For α ∈ NN _{we put}

Bα = {(yn) ∈ c0 : |yn| ≤ sα,n for n ∈ N}.

One can show that (Bα)α∈NN is a strong compactoid

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Sketch of Proof

α = (yα,n) ∈ c0 such that

|xn| ≤ |yα,n| for all x = (xj) ∈ Aα, n ∈ N. Let

sα,n = sup{|xn| : x = (xj) ∈ Aα for n ∈ N}.

Then sα,n ≤ |yα,n| for n ∈ N, so sα,n →n0.

If α, β ∈ NN _{and α ≤ β, then s}

α,n ≤ sβ,n for n ∈ N.

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Sketch of Proof

(105)

Sketch of Proof

(106)

Sketch of Proof

(A2). Now let the field K be arbitrary.

For α ∈ NN _{we put}

(107)

Sketch of Proof

B = {(y ) ∈ c : |y | ≤ s _{for n ∈ N}.}

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Sketch of Proof

(109)

Sketch of Proof

(B). Let (Nk) be a partition of N into infinite subsets and let

φk : N → N be strictly increasing with φk(N) = Nk for k ∈ N.

If α, β ∈ NN _{with α ≤ β, then α ◦ φ}

k ≤ β ◦ φk for all k ∈ N.

The map φ : NN_{→ (N}N₎N _{defined by φ(α) = (α ◦ φ} n) is a bijection. For α ∈ NN _{we put D} α =Q ∞ n=1Bα◦φn.

Using [Gra1, Propositon 1.7] one can prove that (Dα)α∈NN is a

strong compactoid resolution of cN 0.

(C). Let F be a subspace of cN

0. Then the sets

Kα = Dα∩ F , α ∈ NN,

form a strong compactoid resolution of F .

(D). E is a metrizable lcs of countable type, so it is isomorphic to a subspace of cN

0 ([GKPS2, Remark 3.6]).

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Sketch of Proof

The map φ : NN_{→ (N}N₎N _{defined by φ(α) = (α ◦ φ} n) is a bijection. For α ∈ NN _{we put D} α =Q ∞ n=1Bα◦φn.

0. Then the sets

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Sketch of Proof

The map φ : NN _{→ (N}N₎N _{defined by φ(α) = (α ◦ φ} n) is a bijection. For α ∈ NN _{we put D} α =Q ∞ n=1Bα◦φn.

0. Then the sets

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Sketch of Proof

The map φ : NN _{→ (N}N₎N _{defined by φ(α) = (α ◦ φ} n) is a bijection. For α ∈ NN _{we put D} α = Q∞ n=1Bα◦φn.

0. Then the sets

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Sketch of Proof

0. Then the sets

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Sketch of Proof

0. Then the sets

(115)

Sketch of Proof

0. Then the sets

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Sketch of Proof

0. Then the sets

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Remark 21

If the field K is locally compact, then the proof of

Theorem 20 is considerably simpler.

Indeed, in this case E is separable and every compactoid set in E is precompact. Let F be the completion of E .

For a dense sequence (xn) in F and α = (αk) ∈ NN we put

Kα:= ∞ \ k=1 αk [ j =1 B(xj, k−1),

where B(x , r ) denotes the closed ball in F with center x and radius r .

It is not hard to check that (Kα)α∈NN is a strong compact

resolution of F .

Consequently (Kα∩ E )α∈NN is a strong compactoid resolution

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Remark 21

If the field K is locally compact, then the proof of

Theorem 20 is considerably simpler.

Kα:= ∞ \ k=1 αk [ j =1 B(xj, k−1),

resolution of F .

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Remark 21

If the field K is locally compact, then the proof of

Theorem 20 is considerably simpler.

Kα:= ∞ \ k=1 αk [ j =1 B(xj, k−1),

resolution of F .

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Remark 21

If the field K is locally compact, then the proof of

Theorem 20 is considerably simpler.

Kα:= ∞ \ k=1 αk [ j =1 B(xj, k−1),

resolution of F .

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Remark 21

If the field K is locally compact, then the proof of

Theorem 20 is considerably simpler.

Kα:= ∞ \ k=1 αk [ j =1 B(xj, k−1),

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References I

[GKP] N. De Grande-De Kimpe, J. K¸akol and C. Perez-Grcia,

Metrizability of compactoid sets in non-archimedean Hausdorff (LM)-spaces, Ultrametric functional analysis,

Contemporary Math., 319(2003), 99–107.

[GKPS1] N. De Grande-De Kimpe, J. K¸akol, C. Perez-Garcia and W.H. Schikhof,

p-adic locally convex inductive limits, p-adic Functional Analysis (Nijmegen, 1996),

Lecture Notes in Pure and Appl. Math., vol. 192, Marcel Dekker, New York, 1997, pp. 159–222.

Orthogonal sequences in non-archimedean locally convex spaces,

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References II

Orthogonal and Schauder bases in non-archimedean locally convex spaces, p-adic Functional Analysis (Ioannina 2000), Lecture Notes in Pure and Appl. Math., vol. 222, Marcel Dekker, New York, 2001, pp. 103–126.

[GPS] N. De Grande-De Kimpe, C. Perez-Garcia and W.H. Schikhof,

Non-archimedean t-frames and FM-spaces, Canad. Math. Bull., 35(1992), 475–483.

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References III

[Gra2] N. De Grande-De Kimpe,

Nuclear topologies on non-archimedean locally convex spaces, Indag. Mathem., 49(1987), 279–292.

[Kat] A.K. Katsaras,

On compact operators between non-archimedean spaces, Annales Soc. Scientifique Bruxelles, 96(1982), 129–137.

[KS] J. K¸akol and W. ´Sliwa,

On metrizability of compactoid sets in non-archimedean locally convex spaces,

Indag. Mathem., N.S., 19(2008), 563–578.

[MS] S. Mercourakis and E. Stamati,

A new class of weakly K-analytic Banach spaces, Comment. Math. Univ. Carolin, 47(2006), 291–312.

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References IV

[Per1] C. Perez-Garcia,

Semi-Fredholm operators and the Calkin algebra in p-adic analysis I-II,

Bull. Soc. Math. Belg., 42(1990), 69–101.

[Per2] C. Perez-Garcia,

Locally convex spaces over non-archimedean valued fields, in: Ultrametric functional analysis,

Contemporary Math., 319(2003), 251–279.

[Sch1] W.H. Schikhof,

Locally convex spaces over non-spherically complete valued fields I-II,

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References V

[Sch2] W.H. Schikhof,

Topological stability of p-adic compactoids under continuous injections,

Report 8644, Department of Mathematics, University of Nijmegen, The Netherlands, 1986, pp. 1–21.

[Sli] W. ´Sliwa,

On topological classification of non-archimedean Fr´echet spaces,