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G.K. Palei
 
and N.P. Sah / BIBECHANA 7 (2011) 39-43 : BMHSS 

 

 39 

BIBECHANA 
A Multidisciplinary Journal of Science, Technology  

  and Mathematics 

ISSN 2091-0762 (online) 
Journal homepage: http://nepjol.info/index.php/BIBICHANA 

 

 

 

Some results on asymptotically normable Frechet spaces 
 

 G.K. Paleia , N.P. Sahb
∗∗∗∗ 

a
Department of Mathematics, B.N. College, Patna University, Patna, India 

b
Department of Mathematics, M.M.A.M. Campus (Tribhuvan University), Biratnagar, Nepal 

Article history: Received 8 October 2010; Revised 18 November 2010; Accepted 20 November 2010 
 

 

Abstract. 
In this paper, it is shown that the asymptotically normable spaces are the smallest class of Frechet 

spaces which contains the nuclear Kothe spaces with continuous norm, the Banach spaces and is 

closed under ε-tensor products and sub-spaces. Again our main aim will be to construct an 

example of a Kothe space which is Montel, admits a continuous norm, but still is not 

asymptotically normable. 

 

Keywords: Asymptotically normable; Frechet space; Kothe space 

 

 Introduction 

 

 Let E be a Frechet space, ||  ||1 ≤ ||  ||2 ≤ ...... a fundamental system of seminorms on E and 

Uk = { x ∈ E : ||x||k ≤ 1} for every k. E is called asymptotically normable, if there is a ko such that 

for every k ≥ ko there is a p so that the seminorms || ||
ok

 and || ||
k

 define equivalent topologies on 

Up. It is easy to see that in this case || ||
ok

 is in fact a norm. 

 This class of spaces appears in investigations about the structure of Frechet spaces and 

about the behaviour of their operators as a natural counterpart of the class of quasi-normable 

spaces introduced by Grothendieck [1]. While the quasi-normable frechet spaces E are those 

which admit an Ω-type condition [2] and for which there exists a nontrivial Frechet space F with 

Ext
1
 (F, E) = 0 [2], the asymptotically normable Frechet spaces E are those which admit a DN-

type condition [3] and for which thee exists a nontrivial Frechet space F with   Ext
1
 (E, F) = 0. 

Nontrivial here could mean: an infinite dimensional nuclear Kothe space. 

 In [2] it is shown that the quasi-normable spaces are the quotient spaces of standard 

spaces of the form 

: for allλ
 

= ∈ = < + ∞ 
 

∑
j

k
N

j j k j k j, k(A, E) x =(x ) E || x || || x || a  

where E is a Banach space and A = (aj, k) a matrix with O < aj, k ≤ aj, k+1 for all k and 

                                                 
∗
 Corresponding author: Nagendra Pd. Sah, Department of Mathematics, M.M.A.M. Campus(Tribhuvan 

University), Biratnagar, Nepal 



G.K. Palei
 
and N.P. Sah / BIBECHANA 7 (2011) 39-43 : BMHSS 

 

 40 

∞∑ j, k

j j, k+1

a
<+

a
 

 We can show that the asymptotically normable spaces are the subspaces of these standard 

spaces. They are the smallest class of Frechet spaces which contains the nuclear Kothe spaces 

with continuous norm, the Banach spaces and is closed under ε-tensor products and subspaces. 

 

A. We use the standard terminology of the theory of locally convex spaces as in [4]. For the 

theory of Ext & the splitting conditions we refer to [3] & [5]. By    (||  ||k)  we always denote an 

increasing sequence of seminorms defining the topology of a Frechet space E and Uk = {x ∈ E : 

||x||k ≤ 1}. In this section we collect some simple results in order to see the concept of asymptotic 

normability in a proper perspective. 

 Let ϕ : (0, ∞) → (0, ∞, be a strictly increasing function. We say E satisfies (DNϕ) if we 

have a ko with the property that for every k there is a p and C > 0 so that the inequality 

 ||x||k ≤ Cϕ(γ)
ko

|| x || +
1

γ
||x||p 

holds for every x ∈ E and γ > 0 [3]. In the case ϕ(γ) = γ  this condition, denoted by (DN), was 

introduced by D. VOG T in [3] to characterize initially the subspaces of the space (s) of rapidly 

decreasing sequences. Subsequently it was proved by him in [5] that every space of type (DN) is 

isomorphic to a subspace of L∞(I)⊗π (s) for some index set I. 

 Consider also the following condition: there is a ko such that for every   k ≥ ko we have p 

≥ ko with the following property: every sequence in E which is Cauchy with respect to ||  ||p and 

converges to 0 with respect to || ||
ok

even converges to 0 with respect to ||  ||k. 

Proposition 1.1 Every asymptotically normable Frechet space is countably normable. 

 We can obtain the proof from [1], page 176. 

Proposition 1.2 The following are equivalent for a Frechet space E : 

(1) E is asymptotically normable. 

(2) There is ko such that for every k there is a p such that for every ε > 0 we can choose M > 0 

with ||  ||k ≤ M || ||
ok

+ ε||  ||p. 

(3) E has property (DNϕ) for some ϕ. 

Proposition 1.3 A Frechet-Schwartz space is asymptotically normable if and only if it is 

countably normable. 

 The question whether these classes coincide in general we shall answer in the negative in 

the next section. 

 Before we do this let us recall that a Frechet space E is called locally normable [6] if 

there is a ko such that on every bounded set B the topologies induced by E and by || ||
ok

 coincide. 

Obviously every asymptotically normable space is locally normable. On the other hand every 

Frechet-Montel space admitting a continuous norm is locally normable. Hence, in view of 

Proposition 1.3, every Frechet-Schwartz space admitting a continuous norm but not countably 

normable  [7], gives an example of a locally normable space which is not asymptotically 

normable (see also Section 2). 

 

B. In this section we shall consider Kothe sequence spaces. Our main aim will be to construct an 

example of a Kothe space λ (B) which is Montel, admits a continuous norm, but still is not 

asymptotically normable. In particular, λ (B) will be an example of a locally normable Kothe 

space which is not asymptotically normable. 

 By Proposition 1.3, λ (B) cannot be a Schwartz space. In this context we should like to 

mention that the well-known example constructed by Kothe in [4] of a Montel space which is not 

∧ 



G.K. Palei
 
and N.P. Sah / BIBECHANA 7 (2011) 39-43 : BMHSS 

 

 41 

a Schwartz space, is asymptotically normable by the following proposition which is an immediate 

consequence of Proposition 1.2 and gives a characterization of asymptotically normable Kothe 

spaces in terms of the defining matrix. 

Proposition 2.1 A Kothe space λ (A) is asymptotically normable if and only if there is ko such 

that for every k we have p so that for every ε > 0 there is an M > 0 with 

  ≤ ε ∈
oj, k j, k j, p'

a Ma + a j N   

Example 2.2 We define 

    (i j v)
k
  for k ≤ v 

  bi j, v; k =  i
v
 (j v)

k
  for v < k ≤ j + v 

    ik-j (j v)
k
  for j + v < k 

 

Then the Kothe space λ (B) is Montel, admits a continuous norm but is not asymptotically 

normable. 

Proof.: Suppose for some k there is a p such that for every ε > 0 there is M > 0 with 

  bi j, v; k+1 ≤ M bi j, v; k + ε bi j, v; p 

for all (i, j, k) ∈ N
3
. We may assume p > k + 1, set v = k + 1, j = p – v and obtain  

  (i j v)
k+1

 ≤ M (i j v)
k
 + ε i

k+1
 (j v)

p
  

 For ε < (j v)
k+1-p

, we let i tend to infinity and get a contradiction. Hence the Kothe space λ 

(B) is not asymptotically normable by Proposition 1.2. 

 Suppose we fix ko and let I ⊂ N
3
 be an index set such that for all k we have 

  bi j v; k ≤ ρ(k)bi j, v; k 

for some ρ(k) > 0 and (i, j, v) ∈ I. If we show that I must be finite, then we have that λ (B) is a 

Montel space [4]. Since the exponent of i is increasing with k, we have j v ≤ ρ(ko+1) for all (i, j, 

v) ∈ I. For any given jo, vo let k > jo + vo + ko. If (i, jo, vo) ∈ I, we get 

  (≤ ρok- j
i k okk m

o o o o
(j v ) ) i (j v )  

where m is some integer which is strictly less than k – jo in all cases. Hence  {i : (i, jo, vo) ∈ I} is 

finite and so I must also be finite. 

 

C. This section is devoted to the proof of our main theorem 3.3. 

 For this, we state the following lemma without proof: 

Lemma 3.1 Let A = ( ai, k; m) be a doubly indexed matrix and Ak = (ai, k; m +  ai+1, k; m). We assume 

that with suitable sequences i(m) and s(m) ≥ m we have the following: 

(1)  ai, k;k = 1  for all i, k. 

(2)  ai, k; m ≥ ai+1, k;m  for all i, m ≤ k. 

  ai, k; m ≤ ai+1, k;m  for all i ≥ i( m), m > k. 

(3)  ai, k; m = 0  for all m < k. 

(4)  ∞∑ i, k;m

i i, k; s(m)

a
<+

a
 

Then there exists an exact sequence  0 → λ (AK) → λ (A) → ω → 0. 

 We will now show that for a given Kothe-Schwartz space λ (B) there is another such 

space λ (A) and a surjection of λ (A) onto ω whose kernel is isomorphic to λ (B). 

Proposition 3.2 For every Kothe-Schwartz space λ (B) with a continuous norm, there exists a 

Kothe-Schwartz space λ (A) with a continuous norm and an exact sequence 

  0 → λ (B) → λ (A) → ω → 0. 

Further, if λ (B) is nuclear, then λ (A) is also nuclear. 

o 

lim 

i 
sup 

k 



G.K. Palei
 
and N.P. Sah / BIBECHANA 7 (2011) 39-43 : BMHSS 

 

 42 

Proof: We assume without loss of generally bj; 1 > 0 for all j and       
j;m

j;m+

b
=

b 1

0  for all m. 

 We determine inductively disjoint, infinite sets of integers  Mk = {n(i, k) : i = 1, 2, ...} 

with ∪Mk = N such that the following are true: 

(5)  ≥
n(i,k) ;m n(i+1, k) ;m

n(i,k) ;k n(i+1, k) ; k

b b

b b
 for m < k 

(6)  ≤
n(i,k) ;m n(i+1, k) ;m

n(i,k) ;k n(i+1, k) ; k

b b

b b
 for m > k, i ≥ m 

(7)  2−≤ in(i,k) ;m

n(i,k) ;m+1

b

b
 for m ≤ i.  

 For a fixed k, the sequence (n(i, k))i=1,2, ...... can be chosen inductively, starting with n(1, k) 

= min N \
 
 
 

U
k-1

l

l=1

M  and such that N \
 
 
 

U
k

l

l=1

M  remains infinite. We now set  

  
n(i,k) ;m

n(i,k) ; k

b

b
 for m < k 

 ai, k; m = 
n(i-1,k) ;m

n(i-1,k) ;k

b

b
 for m > k, i ≥ 2 

  1 elsewhere 

 Since ai, k; k = ai+1, k; k = 1, we note that (ai, k; m) is increasing in m. Then     (1) ..... (4) of 3.1 

Lemma are satisfied and hence we have an exact sequence          0 → λ (AK) → λ (A) → ω → 0. 

One can check easily that λ (A) is a Schwartz space or if λ (B) is nuclear, then λ (A) is also 

nuclear. 

 For m < k and m ≥ k using (5) and (6) are estimate in different ways and obtain 

 2≤ ≤
n(i,k) ;m n(i,k) ;m

i,k;m i+1,k;m

n(i,k) ;k n(i,k) ;k

b b
a +a

b b
  

for i > m. Therefore λ (AK) is isomorphic to λ (B) by diagonal transformation. 

Theorem 3.3 A Frechet space E is asymptotically normable if and ony if there is an index set I 

and a nuclear Kothe space λ (A) with a continuous norm such that E is isomorphic to a subspace 

of L∞(I)⊗π λ(A). 

Proof: We have a nuclear Kothe space λ (B) with a continuous norm such that     (E, λ (B) ) ∈ 

( )*

1
S  by Lemma 5.4 in [3]. By the previous result we construct an exact sequence. 

  0 → λ (B) → λ (A) → ω → 0. 

 If E is a Banach space, our theorem is trivial. If E is a proper Frechet space, then (E, 

λ (B) ) ∈ ( )
o

*

1
S  by Lemma 3.3 in [5]. Now we choose a set I such that each Banach space Ek is 

isomorphic to a subspace of L∞(I), where Ek is the completion of (E, ||  ||k). We identify the tensor 

product L∞(I)⊗π λ(B) with the space of all y = (yj), such each vj ∈ L∞(I) and for every k ∈ N we 

have  ||y||k = sup ||yj||∞bj;k < + ∞ 

 

where ||  ||∞ denotes the norm of L∞(I). An obvious modification of Proposition 3.5 in [5] yields 

(E, L∞(I)⊗π λ(B) ∈ (S1). Taking tensor product with L∞(I) we get an exact sequence 

lim 

j 

∧ 

∧ 

j 



G.K. Palei
 
and N.P. Sah / BIBECHANA 7 (2011) 39-43 : BMHSS 

 

 43 

 0 → L∞(I)⊗π λ(B) → L∞(I)⊗π λ(A) → L∞(I)
N
 → 0 

 Since Ext (E, L∞(I)⊗π λ(B)) = 0, the natural imbedding T of E into L∞(I)
N 

can be lifted to 

a continuous linear operator T : E → L∞(I)⊗π λ(A). It is easily checked that T is one-to-one and 

has a closed range. 

 We would like to point out that the Kothe space λ(A) we have constructed actually 

depends only on ϕ. This is so, because for each strictly increasing function ϕ : (0, ∞) → (0, ∞), an 

examination of the proof of Lemma 5.4 in [3] reveals that there is a nuclear Kothe space  λ(B) 

such that (E, λ(B)) ∈ ( )*

1
S  for any Frechet space E which satisfies (DNϕ). 

 Finally, we would like to note that L. Holmstrom [8] proved that every nuclear 

asymptotically normable Frechet space is isomorphic to a subspace of some nuclear Kothe space 

which admits a continuous norm. 

 

D. Finally we use the results of section 3 for another description of the class of asymptotically 

normable Frechet spaces and for a comparison with the class quasi-normable Frechet spaces. The 

first Lemma is obvious. 

Lemma 4.1 If E is asymptotically normable and H ⊂ E a subspace then H is asymptotically 

normable. 

Lemma 4.2 If E and F are asymptotically normable then so is 
1

b b
L (E , F) . 

Proof: The proof is straightforward if one notices that E being asymptotically normable is 

equivalent to: there is ko such  that for every k there is a p so that for every ε > 0 we can choose 

M > 0 with 

 ⊂ ε
o

o o o

k k pU MU + U  From these Lemmas and Theorem 3.3, we can obtain: 

Theorem 4.3 The class of asymptotically normable Frechet spaces is the smallest class of Frechet 

spaces containing the nuclear Kothe spaces with continuous norm the Banach spaces and is 

closed under ε-tensor products and subspaces. 

 For comparison we recall that the class of quasi-normable Frechet spaces is the smallest 

class of Frechet spaces containing the nuclear Kothe spaces (or nuclear Kothe spaces with 

continuous norms, or Schwartz spaces, or countably normable Schwartz spaces) and the Banach 

spaces and is closed under π-tensor products and quotient spaces. 

 

References 

 
[1]   A. Grothendieck, Topological vector spaces, New York – London-Paris, (1973). 

[2]   R. Meise, D. Vogt, Math.  Nachr., 122 (1985) 141. 

[3]   D. Vogt, Some results on continuous linear maps between Frechet spaces, in Bierstedt   

        K.D., Fuchssteiner B. (eds.), <<Functional Analysis : Surveys and Recent Results II>>,   

        North-Holland Math. Studies, 90 (1984), pp. 349-381. 

[4]   G. Kothe, Topological vector spaces, I, II, New York-Heildelber-Berlin, 3 (1969, 1979). 

[5]   D. Vogt,  Studia Math., 85 (1987) 163. 

[6]   T. Terzioglu, D. Vogt, Math., 54 (1990)180. 

[7]   E. Dubinsky, Studia   Math., 71 (1981) 85. 

[8]   L. holmstrom, Proc. Amer. Math. Soc., 89 (1983)453. 

Further reading 

 [1]  E. Dubinsky, The structure of nuclear Frechet spaces, Lecture notes in Math., 720  (1979),      

        Berlin-Heildel-berg-New York. 

 [3]  E. Dubinsky, D. Vogt, Frechet spaces with quotients failing the bounded approximation   

        property, Studia Math., 81 (1985), pp.71-77. 

[11] G.L., Litvinov, Nuclear space, Springer Verlag, New York, (2001). 

∧ ∧ 

∧ 

∧