Reports of the Academy of Sciences of the USSR
1968. Volume 180, No. 1

UDC 513.881

MATHEMATICS

B. S. BRUDOVSKII

MAPPINGS OF TYPE \(s\) OF LOCALLY CONVEX SPACES

(Presented by Academician P. S. Novikov on 20 VI 1967)

In the first part of the present note, the concept of mappings of type \(s\) of normed spaces, which were considered by Grothendieck \((^4)\) (under the name of Fredholm mappings of zero order) and by Pietsch \(((^5), 8.5)\), is carried over to locally convex spaces. At the same time, the basic properties of mappings of type \(s\) of normed spaces established in \((^1)\) are preserved also for mappings of type \(s\) of locally convex spaces. In the second part, a number of examples of strongly nuclear function spaces are given.

1. Definition 1. Let \(E\) be a separated locally convex space and let \(E'\) be its dual. A sequence \((x_n)\) in \(E\) (respectively \((x_n')\) in \(E'\)) will be called locally rapidly decreasing if \((x_n)\) (respectively \((x_n')\)) decreases rapidly in the normed space \(E_A\) (respectively \(E_B'\)) for some bounded set \(A\) in \(E\) (respectively some equicontinuous set \(B\) in \(E'\)); a set \(C\) in \(E\) (respectively in \(E'\)) will be called strongly nuclear* if \(C\) is contained in the closed (respectively in the \(\sigma(E',E)\)-closed) absolutely convex hull of some locally rapidly decreasing sequence in \(E\) (respectively in \(E'\)).

Definition 2. Let \(E\) and \(F\) be separated locally convex spaces. A linear mapping \(f:E\to F\) will be called a mapping of type \(s\) if the image of some neighborhood of zero in \(E\) is a strongly nuclear set in \(F\).

In my note \((^1)\), Theorem 3b, it was shown that a mapping \(f\) of a normed space \(X\) into a normed space \(Y\) is a mapping of type \(s\) if and only if the image of the unit ball of \(X\) is a strongly nuclear set in \(Y\). Thus, for the case of normed spaces, the definition given above and the usual definition of mappings of type \(s\) are equivalent.

Let \(E,F\) be separated locally convex spaces and let \(t_{sN}\) be the strongest strongly nuclear topology** in \(E\), majorized by the original one.

Theorem 1. The following conditions for a mapping \(f\in \mathcal L(E,F)\) are equivalent:

a) \(f\) is a mapping of type \(s\);

b) \(f:(E,t_{sN})\to F\) is bounded (i.e., the image of some neighborhood of zero is a bounded set);

c) there exist an equicontinuous sequence \((x_n')\) in \(E'\), a bounded sequence \((y_n)\) in \(F\), and a rapidly decreasing numerical—

* A sequence \((x_n)\) in a locally convex space \(X\) is called rapidly decreasing if \(n^k x_n \to 0\) in \(X\) for every \(k\); a set \(A\subset X\) is called nuclear if \(A\) is contained in the closed absolutely convex hull of some rapidly decreasing sequence from \(X\) (see \((^1)\)).

** A locally convex space \(X\) is called strongly nuclear \((^1)\) if for every neighborhood of zero \(U\) there exists a neighborhood of zero \(V\subset U\) such that the canonical embedding \(X_{U^0}'\to X_{V^0}'\) is a mapping of type \(s\).

a numerical sequence \((\lambda_n)\) such that

\[ f(x)=\sum_N \lambda_n \langle x,x'_n\rangle y_n \quad \text{for all } x\in E; \]

c) there exists a locally rapidly decreasing sequence \((x'_n)\) in \(E'\) such that

\[ p(f(x))\leq \lambda_p \sum_N |\langle x,x'_n\rangle| \]

for every continuous seminorm \(p\) on \(F\) and all \(x\in E\), where \(\lambda_p\) depends only on \(p\);

d) there exist a strongly nuclear set \(M\) in \(E'\) and a positive Radon measure \(\mu\) on \(M\) such that

\[ p(f(x))\leq \lambda_p \int_M |\langle x,x'\rangle|\,d\mu \]

for all \(x\in E\) and every continuous seminorm \(p\) on \(F\), where \(\lambda_p\) depends only on \(p\).

Proposition 1. The mapping adjoint to a mapping of type \(s\) is also a mapping of type \(s\).

A locally convex space \(X\) is called \(\sigma\)-quasi-barreled \((^5)\) if every strongly bounded sequence in \(X'\) is equicontinuous.

Proposition 2. Let \(E\) be \(\sigma\)-quasi-barreled, and let \(F\) be arbitrary locally convex spaces and \(f\in L(E,F)\). If \({}^{t}f\) is a mapping of type \(s\), then \(f\) is also a mapping of type \(s\).

Proposition 3. In a metrizable space every nuclear set is strongly nuclear.

Corollary. Let \(E\) be arbitrary, and let \(F\) be metrizable locally convex spaces. A mapping \(f\in \mathcal{L}(E,F)\) is a mapping of type \(s\) if and only if the image of some neighborhood of zero from \(E\) is a nuclear set in \(F\).

We shall give a characterization of strongly nuclear spaces by means of mappings of type.

Proposition 4. A locally convex space is nuclear if and only if every continuous mapping of it into a Banach space is a mapping of type \(s\).

1. We give some examples of strongly nuclear spaces.

Let \(G\) be an open subset in \(R\times R\), and let \(\mathcal{H}(G)\) be the space of all harmonic functions on \(G\) with the topology of compact convergence.

Theorem 2. The space \(\mathcal{H}(G)\) is strongly nuclear.

Let \((m_{pq})_{(p;q)\in N\times N}\) be a numerical matrix such that \(1\leq m_{0q}\leq m_{1q}\leq \cdots \leq m_{pq}\leq \cdots\) for each \(q\), and let \(k(m_{pq})\) be the set of all numerical sequences \((x_q)\) for which \(\sup_q |m_{pq}x_q|<\infty\) for every \(p\). The space \(k(m_{pq})\) becomes countably normed if one sets \(\|(x_q)\|_p=\sup_q |m_{pq}x_q|\).

Theorem 3. The space \(k(m_{pq})\) is strongly nuclear if and only if, for every \(p_1\), there exists \(p_2\) such that the sequence

\[ \left(\frac{m_{p_1q}}{m_{p_2q}}\right)_q \]

is rapidly decreasing.

Now let \((M_p)_{p\in N}\) be a sequence of continuous functions on the line, increasing at each point: \(1\leq M_0(x)\leq M_1(x)\leq \cdots \leq M_p(x)\leq \cdots\) for each \(x\in R\). By \(\mathcal{K}(M_p)\) (see \((^3)\)) is denoted the space of all infinitely differentiable functions on the число-

line, for which

\[ \max_{0\le r\le p}\sup_{x\in R}\left|M_p(x)\varphi^{(r)}(x)\right|<\infty . \]

This space becomes countably normed if one sets

\[ \|\varphi\|_p=\max_{0\le r\le p}\sup_{x\in R}\left|M_p(x)\varphi^{(r)}(x)\right|. \]

Theorem 4. If for each \(p\in N\) there is a \(q\in N\) such that

\[ \frac{M_p(x)}{M_q(x)}\,|x|^k \longrightarrow 0 \quad \text{as } |x|\to\infty \]

for any \(k\), then the space \(\mathcal K(M_p)\) is strongly nuclear.

In particular, if

\[ M_p(x)=\exp\left[a\left(1-\frac1p\right)|x|^{1/\alpha}\right], \]

where \(a,\alpha>0\) (the space \(\mathcal K(M_p)\) with this sequence \((M_p)\) is denoted by \(S_\alpha\) (see \((^3)\)), then it is easy to see that the sequence \(M_p\) satisfies the condition of Theorem 4. Therefore we have

Corollary. The space \(S_\alpha\) is strongly nuclear.

Moscow State Pedagogical Institute
named after V. I. Lenin

Received
19 VI 1967

REFERENCES

\(^{1}\) B. S. Brudovskii, DAN, 178, No. 2 (1968).
\(^{2}\) N. Bourbaki, Topological Vector Spaces, IL, 1959.
\(^{3}\) I. M. Gelfand, G. E. Shilov, Generalized Functions, vol. 2, Moscow, 1958.
\(^{4}\) A. Grothendieck, Mem. Am. Math. Soc., No. 16 (1955).
\(^{5}\) A. Pietsch, Nuclear Locally Convex Spaces, Moscow, 1967.