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 nonarchimedean functional
analysis. We study some types of limited sets and operators which called
λlimited 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 λlimited spaces and study the relation between λlimited
spaces and λsemiMontel spaces.
PRELIMINARIES
Let K be a field. A nonArchimedean 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 = αβ 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)→λμ
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 nonarchimedean 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 nonarchimedean norm on E. The pair (E, .) is called
a nonarchimedean normed space.
Throughout this study K will stand for a complete non archimedean valued
field, whose valuation is nontrivial. The collection of all continuous
nonarchimedean 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 nonarchimedean 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 nonarchimedean 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 nonarchimedean normed space
for the nonarchimedean norm  _{p} defined by π_{p}(x)_{p}
= p (x), xεE. By De GrandeDe Kimpe and PerezGarcia (1994) is
a space of countable type.
The following sequence spaces will be need:
Definition 1: A nonArchimedean sequence ideal λ on the valued
field K is a subset of the space ι_{∞} (K) satisfying
the following conditions:
• 
e_{i}ελ where e_{i} = (0,0,...,1,....)
the one in the ith place. 
• 
If x_{1}, x_{2}ελ, then x_{1}+x_{2}ελ. 
• 
If yει_{∞} (K) and xελ,
then x.y ε λ. 
• 
If the sequence x = (x_{0},x_{1},.....)ελ,
then (x_{0},x_{0}, x_{1},x_{1},...)ελ.

Note that the sequence spaces, ι_{∞} (K), c_{0}
(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 nonnegative 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 PerezGarcia, 1997). These nth Kolmogrove diameters
satisfy the following properties:
Proposition 1:
• 
δ_{0,p}(B)≥δ_{1,p}(B)≥δ_{2,q}(B)≥......≥0.
for all pεcs (E). 
• 
If B_{1}⊆B and p≤q, then δ_{n,p}(B_{1})≤δ_{n,q}
(B). 
• 
If TεL (E, F), then for all pεcs(F) there exists
qεcs (E) such that δ_{n,p} (T(B))≤δ_{n,q}
(B). 
• 
If p′≥p, then δ_{n} (π_{p} (B_{q}(0,1)),
π_{p}(B_{p′}(0,1))) = δ_{n}
(B_{q} (0,1), B_{p} (0,1)) (Dubinsky, 1979; Jarchow,
1981; Safi, 2006). 
λLIMITED SETS
Definition 2: Let E, F be locally convex spaces over K, then
• 
A subset B of E is called compactoid if for every zeroneighborhood
U in E there exists a finite set A⊂E such that B⊂co (A)+U,
where co (A) is the absolutely convex hull of A. 
An operator TεL (E, F) is called compactoid if there exists a
zeroneighborhood U in E such that T (U) is compactoid in F (De GrandeDe
Kimpe et al., 1995). Katasars and C. PerezGarcia (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 (S_{n,p})εc_{0}
(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 c_{0} (k), T (B) is
compactoid in c_{o} (K). 
An operator TεL (E, F) is called limited if there exists a zeroneighborhood
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 GrandeDe Kimpe
et al., 1995).
Parallel to this definitions we define the following:
Definition 3: Let E, F be locally convex spaces over K, then
• 
A subset B of E is called_{ }λcompactoid
if we replace c_{o} (K) in (1) 
• 
by the sequence ideal. 
An operator TεL (E, F) is called λcompactoid if there exists
a zeroneighborhood U in E such that T (U) is λcompactoid in F. We will denoted by λC (E,F) the space of
all λcompactoid operators from E into F.
• 
A bounded subset B of E is called λlimited in
E if and only if for each continuous linear map T from E to c_{o}
(K), T (B) is λcompactoid in c_{o} (K). 
An operator TεL(E,F) is called λlimited if there exists
a zeroneighborhood U in E such that T (U) is λlimited in F. We
will denoted by (λ)lim (E, F) the space of all λlimited operators
from E into F.
Notes:
• 
If dim (E) = n, then every bounded subset of E is λcompactoid. 
• 
If 
are two subsets of ι_{1} (K), then according to Pietch
(1972) we have
and
where B_{ι1} is the closed unit ball in ι_{1}.
Hence D is c_{0} (K) compactoid and (S)compactoid, but not (R)compactoid
and B is c_{0} (K)compactoid but not (S)compactoid.
Proposition 2: Let E,F be locally convex spaces over K, then
i) 
Every λcompactoid subset of E is λlimited
in E. 
ii) 
If B is λlimited in E and TεL (E, F), then T (B) is
λlimited in F. 
iii) 
If B is λlimited in E and D⊂B, then D is λlimited
in E. 
iv) 
If A is λlimited, then 
v) 
If A, B⊂E are λlimited in E, then A+B is λlimited
in E. 
vi) 
The product of any finite number of λlimited sets is λlimited. 
vii) 
Let M be a subspace of E and B⊂M. If B is λlimited in
M, then B is λlimited in E (DeGrandeDe Kimpe et al.,
1995). 
Proof:
• 
Let B be any λcompactoid subset of E and let TεL
(E, c_{0}(K)). It follows from property (iii) of proposition
(1) that for all pεcs(F) there exists qεcs (E) such
that δ_{n, p} (T (B))≤δ_{n.q}(B) and
so T (B) is λcompactoid in c_{0} (K). Therefore B is
λlimited in E. 
• 
Suppose B is λlimited in E and TεL (E, F). Let GεL
(F,c_{0} (K)), then G_{o}TεL (E, c_{0}(K)).
It follows that G (T(B)) is λcompactoid in c_{0}(K)
and so T(B) is λlimited 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 δ_{n,
p} (T(D))≤δ_{n, p} for all pεcs (F).
Since B is λlimited in E, then T (D) is λcompactoid in
c_{0} (K) and this complete the proof. 
• 
From definition of δ_{n,p} (A), if ε>0 there
exist a subspace F of E with dim(F)≤n and μεK such
that μ≤δ_{n,p}(A)+ε, A⊆μB_{p}(0,1)+F.
It follows thatĀ⊆μB_{p} (0,1)+F and so δ_{n,p}
(A)≤μ≤δ_{n,p}(A)+ε. Since ε>0
is an arbitrary, we deduce that δ_{n,p} (Ā)≤
δ_{n,p} (A). That is, if A is λcompactoid in E,
then _{0}(K)). Since A is λlimited it follows that T(A)
is λcompactoid and hence is
λcompactoid. Since,
it follows that T(Ā) is λcompactoid. 
• 
Let TεL (E,c_{0}) Since A, B are λlimited,
then T(A), T(B) are λcompactoid. Since, T(A+B)⊆T(A)+T(B),
it follows by Safi (2006) that T (A+B) is λcompactoid in c_{0}(K)
and so A+B is λlimited in E. 
• 
Let D_{i} be any λlimited set in E_{i}, i
= 1,2,......, n and let E = E_{1}xE_{2}x.....xE_{n},
TεL(E, c_{0}(K)). Now If π_{i}: E_{i}→E
is conical inclusion, then the operator T_{i} = T○π_{i}εL(E_{i},c_{o}(K))
and so T_{i} (D_{i}) is λcompactoid in c_{0}(K).
Since, 
then
is λcompactiod (Safi, 2006). Therefore
is λlimited (proposition 2.i).
• 
Let M be a subspace of E and let B be λlimited
M. If TεL (E, F), then the restriction operator TMεL
(M, c_{0}(K)). Since TMεL(M,c_{o}(K)) is
λcompactoid in c_{0} (K) it follows that B is λlimited
in E. 
Note: If λ = c_{0} (K), then the unit ball ι_{∞}
of is λlimited, but not λcompactoid (De GrandeDekimpe and
PerezGarcia, 1994).
Definition 4: A locally convex space over K is called λGelfandPhilips
space (λGPspace in short) if every λlimited set in E is λcompactoid.
(De GrandeDekimpe and PerezGarcia, 1994).
Remark : c_{0} (K) is λGp space, for any sequence
ideal λ (and hence any nonarchimedean normed space of countable
type (Van Rooij, 1978).
To see that let A be any λlimited set in c_{0} (K). Since
the identity operator IεL (c_{0}(K), c_{0}(K)),
then I(A) = A is λcompactoid.
λLIMITED SPACES
De Ggrandede Kimpe and PerezGaria (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 nonarchimedean normed space F.
Definition 5: We say that the locally convex space E over K is
λlimited space if L (E, F) = λlim (E,F) for all nonarchimedean
normed spaces F.
Notes:
• 
If λ = c_{0}(K), then the concepts of λlimited
space coincide with the limited spaces and if the valuation K is dense.
Then ι_{∞}(K) is λlimited spaces.
Since L(c_{0} (K), c_{0} (K))≠λC(c_{0}
(K), c_{0} (K))⊆λlim (c_{0} (K), c_{0}
(K)), then c_{0} (K) is not λlimited spaces (De GrandeDe
Kimpe et al., 1995). 
• 
If E is a nonarchimedean normed space, then the closed unit ball
of E, B_{E} is λlimited if L (E, c_{0} (K))
= λC (E, c_{0} (K)). 
Theorem 1: If L (E, F) = λlim (E, F) for any locally convex
spaces E, F over K and M is a closed subspace of E then. L (E/M, F) =
λlim (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) = λlim (E, F), there exists a zeroneighbourhood
U in E such that (T○π(U) = T (π(U)) is λlimited.
Since π (U) is a zeroneighbourhood in E/M, then Tελlim
(E/M, F).
Proposition 3: Let F, E_{1}, E_{2}, ..... be any
locally convex spaces over K:
• 
If L (E_{i}, F) = λlim (E_{i},
F) for each iεN, then. 
• 
If L (F, E_{i}) = λlim (F, E_{i})
for each iεI, I is finite, then 
• 
If E_{i} is λGPspace and L (F,
E_{i}) = λlim (F, E_{i}) for each iεN,
then 
(Van Rooij, 1978).
Proof:
and let Tε(E, F). Then T is bounded on some zeroneighborhood
W of E. This neighborhood can be taken in the form
where U_{i} is a zeroneighborhood in E_{i} and the set
J = {iεN: U_{i}≠E_{i}} is finite. So we can
assume that E = E_{1}xE_{2}x.....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_{i}εL
(E_{i}, F) and L (E_{i}, F) = λlim (E_{i},
F), then there exists a zero neighborhood V_{i} in E_{i}
such that T_{i} (V_{i}) is λlimited set in F, then
V = VxV_{2}x.....xV_{n} is zeroneighborhood in E for
which T(V) = T_{1}(V_{1})+T_{2}(V_{2})+.....+T_{n}
(V_{n}) is λlimited set in F (proposition (2.v)). So. Tελlim
(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}) = λlim (F,E_{i}), then P_{i}○T
is λlimited operator. Thus, there exists a zeroneighborhood U in
F such that P_{i}○T (U) = W_{i} is λlimited
set in E_{i}. It follows by proposition (2.vi)
is λlimited set in
and so T is λlimited operator.
then like in part (ii) we can find a zeroneighborhood U in F such that
P_{i}○T (U) = W_{i} is λlimited set in E_{i}
for all iεN. Since E_{i} is λGPspace, then W_{i}
is λcompactiod set in E_{i}. Now by Safi (2006)
is λcompactod set and by proposition (2.i) T(U) is λlimited
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 λlimited space.
Proof: Sufficiency, let E be λlimited space and let pεcs(E).
Since E_{p} = E/Ker p is a nonarchimedean 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 λlimited in E_{p}. Now since E_{p}
is a nonarchimedean normed space of countable type, then E_{p}
is λGpspace and so π_{p} (B_{q} (0, 1) is
λcompactoid 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 nonarchimedean 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 λcompactoid in F and by proposition
(2.i) is λlimited, Thus T is λlimited operator.
Definition 7: A locally convex space E over K is called λsemiMontel,
if every bounded subset D of E is λcompactoid.
Notes:
• 
Every finite dimensional normed space is λsemiMontel. 
• 
If E is λGp space such that every bounded subset of E is λlimited,
then E is λsemiMontel space. 
• 
If EλsemiMontel space, then every bounded subset of E is
λlimited. 
Proposition 5:
• 
If E is λlimited space, then every bounded set
in E is λlimited. 
• 
If F is a locally convex space over K and L (E, F) = λlim
(E, F) for every nonarchimedean normed space E, then every bounded
set in F is λlimited. 
• 
If F is λsemiMontel space, then L (E, F) = λlim (E,
F) for every nonarchimedean normed space E. 
• 
If F is λGp space and L (E, F) = λlim (E, F) for every
nonarchimedean normed space E, then F is λsemiMontel space. 
Proof:
• 
Let A be any bounded subset of E and let TεL
(E, c_{0} (k)). Since L (E,c_{o} (K)) = λlim
(E, c_{o} (K)), then there exists a zeroneighborhood U in
E such that T(U) is λlimited in c_{0}(K). Since A is
bounded, then there exists rεK, r>0, such that A⊂rU
and so T(A)⊂rT(U). It follows by proposition (2.iii) T(A) is λlimited
in c_{0}(K). Since c_{0}(K) is λGp space, then
T(A) is λcompactoid and so A is λlimited set in E. 
• 
Suppose_{ }L (E, F) = λlim (E, F) for every nonarchimedean
normed space E. We shall show that every bounded set A in F is λlimited.
Since A is bounded set in F, then for each pεcs (F) there
exists m(p)εK, m(p)>0 such that A⊂m(p) B_{p}
(0, 1). Now let, 
Then q is a nonarchimedean seminorm on A. If q(y) = 0, then p (y) =
0 for all pεs(F) and so y = 0. Thus q is a nonarchimedean norm.
Now by E, we shall take the nonarchimedean 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) = λlim (E, F), then T is λlimited operator.
Thus T (B_{E}) = B_{E} is λlimited set in F and
by proposition (2. iii) A is λlimited set in F.
• 
Let F be any λsemiMontel space, E be a nonarchimedean
normed space and TεL (E, F). Since T is bounded, then T maps
the unit ball B_{E} into E a bounded set T(B_{E})
in F and so T (B_{E}) is λcompactoid set in F, hence
T(B_{E}) is λlimited set in F. Therefore T is λlimited
operator. 
• 
It follows from part (2) and the fact that the space F is λGp
spaces 
Theorem 2: Let F, E be any locally convex spaces over K and let
L (E, F) = λlim (E, F), then L (E_{0}, F_{0}) =
λlim (E_{0}, F_{0}) for a complement linear subspace
E_{0} of E and subspace F_{0} of F.
Proof: Let T_{0}εL (E_{0}, F_{0})
and let TεL (E, F) defined by T(x) = T_{0} where x = x_{0}+x_{1},
x_{0}εE_{0}. Since L (E, F) = λlim (E, F),
then T is λlimited operator and so there exists a zeroneighborhood
U in E such that T(U) is λlimited set. Since U∩E_{0}
is zeroneighborhood in E_{0}, then applying proposition (2.iii)
we deduce that T (U∩E_{0}) = T_{0} (U∩E_{0})
is λlimited and therefore T_{0} is λlimited operator.
Note: If the valuation on K is dense and λ = c_{0}
(K), then L (ι_{∞} (K), ι_{∞}
(K)) = λlim (ι_{∞} (K), ι_{∞
}(K)) but L (c_{0} (K), c_{0}(K)) ≠ λlim
(c_{o} (K), c_{o} (K)) (De GrandeDe Kimpe et al.,
1995, Example (2.6.iv)).