INTRODUCTION

In this research we study locally convex spaces over a complete valued scalar field K that are not isomorphic to R or C.We treat some theorems about compact sets and operators in functional analysis over R or C and discussion whether or not they remain valid in non-archimedean functional analysis. We study some types of limited sets and operators which called λ-limited sets and operators in non- archimedean locally convex spaces and we use the Kolmogrove diameters to obtain results resembling previously known properties of limited sets. We generate the concept of limited spaces into λ-limited spaces and study the relation between λ-limited spaces and λ-semi-Montel spaces.

PRELIMINARIES

Let K be a field. A non-Archimedean valuation on K is a function |.|:K→[0,∞) such that for all α,βεK it satisfies: |α| = 0 if and only if α = 0; |αβ| = |α||β| and |α+β|≤max{|α|,|β|}. Note that the last condition separates the absolute value on R or C from all other valuations. The mapping (λ,μ)→|λ-μ| is a metric on K making K into a topological field. We will call the valuation is dense if the set |K|\{0}, where |K| = {|λ|:λεK}, is dense in (0,∞).

Let E be a vector space over the field K. A non-archimedean seminorm on E is a seminorm which verifies the strong triangle inequality: ||a+b||≤max{||a,b||} for all a,bεE. If in addition ||x|| = 0⇔x = 0, then we say that ||.|| is a non-archimedean norm on E. The pair (E, ||.||) is called a non-archimedean normed space.

Throughout this study K will stand for a complete non- archimedean valued field, whose valuation is non-trivial. The collection of all continuous non-archimedean seminorm on a vector space E over K will be denoted by cs (E). For pεcs (E) and r>0, B_p (0, r) will be the set {xεE: p (x)≤r}. L (E, F) will be the vector space of all continuous linear operators from E into F. The non-archimedean normed space E is said to be of countable type if, there exists a countable subset S of E, such that the subspace [S] spanned by S is dense in E.

For a continuous non-archimedean seminorms p on E we put E_p = E/ker p and denoted by π_p the canonical surjection π_p: E→E_p. Then E_p is a non-archimedean normed space for the non-archimedean norm || ||_p defined by ||π_p(x)||_p = p (x), xεE. By De Grande-De Kimpe and Perez-Garcia (1994) is a space of countable type.

The following sequence spaces will be need:

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Definition 1: A non-Archimedean sequence ideal λ on the valued field K is a subset of the space ι_∞ (K) satisfying the following conditions:

Note that the sequence spaces, ι_∞ (K), c₀ (K), (S), (R) and Λ (α) are examples of sequence ideals (Pietch, 1980).

For a bounded subset B of a locally convex space E over K, a pεcs (E) and a non-negative integer n, the nth Kolmogrove diameter δ_n,p(B) (or δ_n(B, B_p (0,1))) of B with respect to p is the infimum of all |μ|, μεK, for which there exists a subspace F of E with dim (F)≤n, such that A⊂F+μB_p (0,1) (Katasars and Perez-Garcia, 1997). These nth Kolmogrove diameters satisfy the following properties:

Proposition 1:

λ-LIMITED SETS

Definition 2: Let E, F be locally convex spaces over K, then

An operator TεL (E, F) is called compactoid if there exists a zero-neighborhood U in E such that T (U) is compactoid in F (De Grande-De Kimpe et al., 1995). Katasars and C. Perez-Garcia (1997) used the Kolmogrove diameters to give the following equivalent definition:

•	The bounded subset B of a locally convex space E over K is called compactoid if and only if (Sn,p)εc0 (K) for all pεcs (E).(1)
•	A bounded subset B of E is called limited in E if and only if for each continuous linear map T from E to c0 (k), T (B) is compactoid in co (K).

An operator TεL (E, F) is called limited if there exists a zero-neighborhood E in U such that T (U) is limited in F. We will denote by lim (E, F) the vector space of all limited operators from E to F ((De Grande-De Kimpe et al., 1995).

Parallel to this definitions we define the following:

Definition 3: Let E, F be locally convex spaces over K, then

An operator TεL (E, F) is called λ-compactoid if there exists a zero-neighborhood U in E such that T (U) is

λ-compactoid in F. We will denoted by λ-C (E,F) the space of all λ-compactoid operators from E into F.

An operator TεL(E,F) is called λ-limited if there exists a zero-neighborhood U in E such that T (U) is λ-limited in F. We will denoted by (λ)-lim (E, F) the space of all λ-limited operators from E into F.

Notes:

If dim (E) = n, then every bounded subset of E is λ-compactoid.

are two subsets of ι₁ (K), then according to Pietch (1972) we have

and

where B_ι1 is the closed unit ball in ι₁. Hence D is c₀ (K)- compactoid and (S)-compactoid, but not (R)-compactoid and B is c₀ (K)-compactoid but not (S)-compactoid.

Proposition 2: Let E,F be locally convex spaces over K, then

Proof:

•	Let B be any λ-compactoid subset of E and let TεL (E, c0(K)). It follows from property (iii) of proposition (1) that for all pεcs(F) there exists qεcs (E) such that δn, p (T (B))≤δn.q(B) and so T (B) is λ-compactoid in c0 (K). Therefore B is λ-limited in E.
•	Suppose B is λ-limited in E and TεL (E, F). Let GεL (F,c0 (K)), then GoTεL (E, c0(K)). It follows that G (T(B)) is λ-compactoid in c0(K) and so T(B) is λ-limited in E.
•	Let D⊂B and let TεL (E, F). Since T(D)⊂T (B), then by property (ii), (iii) of proposition (1) it follows that δn, p (T(D))≤δn, p for all pεcs (F). Since B is λ-limited in E, then T (D) is λ-compactoid in c0 (K) and this complete the proof.
•	From definition of δn,p (A), if ε>0 there exist a subspace F of E with dim(F)≤n and μεK such that |μ|≤δn,p(A)+ε, A⊆μBp(0,1)+F. It follows thatĀ⊆μBp (0,1)+F and so δn,p (A)≤|μ|≤δn,p(A)+ε. Since ε>0 is an arbitrary, we deduce that δn,p (Ā)≤ δn,p (A). That is, if A is λ-compactoid in E, then 0(K)). Since A is λ-limited it follows that T(A) is λ-compactoid and hence is λ-compactoid. Since, it follows that T(Ā) is λ-compactoid.
•	Let TεL (E,c0) Since A, B are λ-limited, then T(A), T(B) are λ-compactoid. Since, T(A+B)⊆T(A)+T(B), it follows by Safi (2006) that T (A+B) is λ-compactoid in c0(K) and so A+B is λ-limited in E.
•	Let Di be any λ-limited set in Ei, i = 1,2,......, n and let E = E1xE2x.....xEn, TεL(E, c0(K)). Now If πi: Ei→E is conical inclusion, then the operator Ti = T○πiεL(Ei,co(K)) and so Ti (Di) is λ-compactoid in c0(K). Since,

then

is λ-compactiod (Safi, 2006). Therefore

is λ-limited (proposition 2.i).

Note: If λ = c₀ (K), then the unit ball ι_∞ of is λ-limited, but not λ-compactoid (De Grande-Dekimpe and Perez-Garcia, 1994).

Definition 4: A locally convex space over K is called λ-Gelfand-Philips space (λ-GP-space in short) if every λ-limited set in E is λ-compactoid. (De Grande-Dekimpe and Perez-Garcia, 1994).

Remark : c₀ (K) is λ-Gp space, for any sequence ideal λ (and hence any non-archimedean normed space of countable type (Van Rooij, 1978).

To see that let A be any λ-limited set in c₀ (K). Since the identity operator IεL (c₀(K), c₀(K)), then I(A) = A is λ-compactoid.

λ-LIMITED SPACES

De Ggrande-de Kimpe and Perez-Garia (1994) give the following definition:

The locally convex space E over K is called limited space if L (E, F) = lim (E, F) for all non-archimedean normed space F.

Definition 5: We say that the locally convex space E over K is λ-limited space if L (E, F) = λ-lim (E,F) for all non-archimedean normed spaces F.

Notes:

Theorem 1: If L (E, F) = λ-lim (E, F) for any locally convex spaces E, F over K and M is a closed subspace of E then. L (E/M, F) = λ-lim (E/M, F).

Proof: Let M be a closed subspace of E and TεL (E/M, F). If π: E→E/M is the quotient map, then T○πεL (E, F). Since L (E, F) = λ-lim (E, F), there exists a zero-neighbourhood U in E such that (T○π(U) = T (π(U)) is λ-limited. Since π (U) is a zero-neighbourhood in E/M, then Tελ-lim (E/M, F).

Proposition 3: Let F, E₁, E₂, ..... be any locally convex spaces over K:

(Van Rooij, 1978).

Proof:

and let Tε(E, F). Then T is bounded on some zero-neighborhood W of E. This neighborhood can be taken in the form

where U_i is a zero-neighborhood in E_i and the set J = {iεN: U_i≠E_i} is finite. So we can assume that E = E₁xE₂x.....xE_n for some nεN. Now for i = 1, 2, ...., n, let π_i: E_i→E be the conical inclusion. Since the operator T_i = T○π_iεL (E_i, F) and L (E_i, F) = λ-lim (E_i, F), then there exists a zero- neighborhood V_i in E_i such that T_i (V_i) is λ-limited set in F, then V = VxV₂x.....xV_n is zero-neighborhood in E for which T(V) = T₁(V₁)+T₂(V₂)+.....+T_n (V_n) is λ-limited set in F (proposition (2.v)). So. Tελ-lim (E, F).

I, is finite, and let P_i: E→E_i be the canonical operator, then P_i○TεL (F, E_i). Since L (F, E_i) = λ-lim (F,E_i), then P_i○T is λ-limited operator. Thus, there exists a zero-neighborhood U in F such that P_i○T (U) = W_i is λ-limited set in E_i. It follows by proposition (2.vi)

is λ-limited set in

and so T is λ-limited operator.

then like in part (ii) we can find a zero-neighborhood U in F such that P_i○T (U) = W_i is λ-limited set in E_i for all iεN. Since E_i is λ-GP-space, then W_i is λ-compactiod set in E_i. Now by Safi (2006)

is λ-compactod set and by proposition (2.i) T(U) is λ-limited set in

Therefore

Definition 6: A locally convex space E over K is said to be of type (S_λ) if for each Pεcs(E) there exists qεcs(E) such that

for each q′≥p) (Zahriuita, 1973).

Proposition 4: The space E is of type (S_λ) if and only if E is λ-limited space.

Proof: Sufficiency, let E be λ-limited space and let pεcs(E). Since E_p = E/Ker p is a non-archimedean normed space and the canonical surjection

is continuous, then π_p is λ- limited operator, so there exists a neighborhood B_q (0, 1) in E such that π_p (B_q (0, 1) is λ-limited in E_p. Now since E_p is a non-archimedean normed space of countable type, then E_p is λ-Gp-space and so π_p (B_q (0, 1) is λ-compactoid set in E_p, hence

for each hεcs (E). Now if p′≥p, then

and by proposition (1. iv) it follows that

thus E is a space of type (S_λ).

Necessity: Let E be a space of type (S_λ), F be an arbitrary non-archimedean normed space and TεL (E, F). Now for the closed unit ball B_F, there exists pεcs (E) such that T (B_P (0,1))⊂B_F. Since E is a space of type (S_λ), there exists qεcs (E) such that

for all p′≥q. It follows by proposition (1. (iii) ) that

for all p′≥p. Now since,T (B_P’ (0,1))⊂T (B_q (0,1))≤ BF)) then (δ_n(T(B_q (0,1)), B_F)) ≤δ (T(B_q (0,1)), T(B_q (0,1)), T (B_p’ (0,1)).

Therefore (δ_n (T (B_q (0,1)), B_F))ελ and so T(B_q (0, 1) is λ-compactoid in F and by proposition (2.i) is λ-limited, Thus T is λ-limited operator.

Definition 7: A locally convex space E over K is called λ-semi-Montel, if every bounded subset D of E is λ-compactoid.

Notes:

Proposition 5:

Proof:

Then q is a non-archimedean seminorm on A. If q(y) = 0, then p (y) = 0 for all pεs(F) and so y = 0. Thus q is a non-archimedean norm. Now by E, we shall take the non-archimedean normed space of all yεF with q (y)<∞. If B_E = {yεF: q(y)≤1} is the closed unit ball of E, then A⊆B_E and if the operator T equal to the identity imbedding of E into F, then TεL (E, F). Since L (E, F) = λ-lim (E, F), then T is λ-limited operator. Thus T (B_E) = B_E is λ-limited set in F and by proposition (2. iii) A is λ-limited set in F.

Theorem 2: Let F, E be any locally convex spaces over K and let L (E, F) = λ-lim (E, F), then L (E₀, F₀) = λ-lim (E₀, F₀) for a complement linear subspace E₀ of E and subspace F₀ of F.

Proof: Let T₀εL (E₀, F₀) and let TεL (E, F) defined by T(x) = T₀ where x = x₀+x₁, x₀εE₀. Since L (E, F) = λ-lim (E, F), then T is λ-limited operator and so there exists a zero-neighborhood U in E such that T(U) is λ-limited set. Since U∩E₀ is zero-neighborhood in E₀, then applying proposition (2.iii) we deduce that T (U∩E₀) = T₀ (U∩E₀) is λ-limited and therefore T₀ is λ-limited operator.

Note: If the valuation on K is dense and λ = c₀ (K), then L (ι_∞ (K), ι_∞ (K)) = λ-lim (ι_∞ (K), ι_∞ (K)) but L (c₀ (K), c₀(K)) ≠ λ-lim (c_o (K), c_o (K)) (De Grande-De Kimpe et al., 1995, Example (2.6.iv)).