MATHEMATICS

B. M. MAKAROV

ON INDUCTIVE LIMITS OF NORMED SPACES

(Presented by Academician V. I. Smirnov, 23 XII 1957)

In the works of Sebastião e Silva (¹) and D. A. Raikov (²), inductive and projective limits of normed spaces with completely continuous embeddings are considered. The present note is devoted mainly to inductive and projective limits of reflexive normed spaces. The previously obtained results (¹,²) carry over, in the main, to the case under consideration.

Definition 1. Let \(E_n\) be a sequence of locally convex spaces, where \(E_n\) is a linear subspace of \(E_{n+1}\) and the embedding (identification operation) of \(E_n\) into \(E_{n+1}\) is continuous. The inductive limit of the sequence of spaces \(E_n\) is their union

\[ E=\bigcup_{1}^{\infty} E_n, \]

endowed with the strongest of the locally convex topologies for which all the embeddings of \(E_n\) into \(E\) are continuous.

Definition 2. Let \(E_n\) be the inductive limit of the spaces \(E_n\), and suppose there exists a sequence of absolutely convex * closed neighborhoods of zero \(V_n\) (\(V_n\) is a neighborhood of zero in \(E_n\)) such that, for some \(\lambda_n>0\), \(\lambda_n V_n \subset V_{n+1}\), \(n=1,2,\ldots\). We shall say that the sequence of spaces \(E_n\) satisfies condition \((F_1)\) if all \(V_n\) are closed in \(E\), while \(V_1\) contains no nonzero linear subspace of \(E\); and, respectively, condition \((F_2)\), if every absolutely convex closed neighborhood of zero \(W_n\) in \(E_n\) that is contained in \(V_n\) is closed in \(E_{n+1}\).

Theorems 1 and 2 are proved under the assumption that one of the conditions \((F_1)\) or \((F_2)\) is satisfied.

Theorem 1. Let \(E\) be the inductive limit of separable spaces \(E_n\). Then \(E\) is separable, and every set bounded in \(E\) is contained in some \(E_n\).

Theorem 2. Let \(E\) be the inductive limit of a sequence of normed spaces \(E_n\). Then every set bounded in \(E\) is contained and bounded in some \(E_n\).

A locally convex space \(E\) is called bornological if every convex set that absorbs every bounded set is a neighborhood of zero in \(E\).

Theorem 3. In order that a separable bornological space have a metrizable conjugate, it is necessary and sufficient that it be the inductive limit of a sequence of normed spaces satisfying condition \((F_1)\).

* A subset \(A\) of a linear set \(L\) is called absolutely convex if, for any \(x,y\in A\) and \(|\alpha|+|\beta|\le 1\), one has \(\alpha x+\beta y\in A\).

Since the inductive limit of bornological (in particular normed) spaces is again a bornological space, from Theorems 2 and 3 we obtain:

Corollary. If \(E\) is the inductive limit of a sequence of normed spaces satisfying condition \((F_2)\), then \(E\) is also the inductive limit of a sequence of normed spaces satisfying condition \((F_1)\).

With the help of Grothendieck’s results \((^3)\), we obtain that every reflexive space with a metrizable dual is the inductive limit of a sequence of normed spaces satisfying condition \((F_1)\).

Definition 3. Let \(E_n\) be a sequence of locally convex spaces, and let \(\pi_m^n\), for all \(m\) and \(n>m\), be continuous linear mappings of \(E_n\) into \(E_m\) such that \(\pi_m^p=\pi_m^n\pi_n^p\) for \(m<n<p\). Let \(E\) be the set of sequences \(x=\{x_n\}\), where each \(x_n\in E_n\) and \(x_m=\pi_m^n(x_n)\) for all \(m\) and \(n>m\), and let \(\pi_n\) be the linear mapping (projection) of \(E\) into \(E_n\) (\(\pi_n(x)=x_n\)). The projective limit of the sequence of spaces \(E_n\) with respect to the mappings \(\pi_m^n\) is the space \(E\), endowed with the weakest topology for which all projections \(\pi_n\) are continuous.

We shall call inductive and projective limits of reflexive normed spaces, respectively, spaces of classes \((J)\) and \((P)\).

It is clear that in this case both conditions \((F_1)\) and \((F_2)\) are satisfied.

Theorem 4. Spaces of classes \((J)\) and \((P)\) are reflexive, and their strong duals are, respectively, spaces of classes \((P)\) and \((J)\).

We agree to denote the weak topologies in a locally convex space \(E\) and in the dual space \(E'\), respectively, by \(\sigma(E,E')\) and \(\sigma(E',E)\).

Theorem 5. Let \(E\) be the inductive limit of reflexive normed spaces \(E_n\), \(A\subset E\), and let \(A_n=A\cap E_n\) be closed in the topology \(\sigma(E_n,E_n')\). Then \(A\) is closed in \(E\) (in the strong topology).

Remark. If \(A\) is a convex set, then from the closedness of \(A_n\) in \(E_n\) (in the strong topology) it follows that \(A\) is closed in \(E\) in the topology \(\sigma(E,E')\).

The last remark somewhat strengthens a known result \((^6)\) (which, however, concerns a broader class of spaces) stating that in a space \(F'\) dual to a complete locally convex metrizable space \(F\), for a convex set \(A\subset F'\) to be closed in the topology \(\rho(F',F)\), it is sufficient that all intersections of \(A\) with absolutely convex weakly closed bounded sets be weakly closed.

For quotient spaces and subspaces of classes \((P)\) and \((J)\), the duality established by D. A. Raikov \((^2)\) in the case of inductive and projective limits with completely continuous embeddings is preserved to a considerable extent.

Theorem 6. Let \(F\) be a space of class \((P)\), and let \(M\) be a closed subspace of \(F\). The quotient space \(F/M\) is again a space of class \((P)\).

Theorem 7. Let \(E\) be a space of class \((J)\), and let \(N\) be a closed subspace of \(E\). The quotient space \(E/N\) is again a space of class \((J)\).

Theorem 8. A closed subspace \(M\) of a space \(F\) of class \((P)\) is again a space of class \((P)\), isomorphic to the strong dual of the space \(F'/M^0\).

We shall say that a linear mapping of one locally convex space into another is weakly completely continuous if the closure of the image of some neighborhood is weakly bicompact.

Let \((J_1)\) be the class of inductive limits of normed spaces with weakly continuous embeddings, and let \((J_1')\) be the class of conjugate spaces.

The spaces of class \((J_1)\) are reflexive, and for spaces \((J_1')\) and \((J_1)\) Theorems 2 and 5–8 hold.

It is easy to prove that the inductive limit of arbitrary separable locally convex spaces with weakly completely continuous embeddings is a space \((J_1)\).

The author expresses deep gratitude to D. A. Raikov and G. P. Akilov for their comments and advice.

REFERENCES

¹ J. Sebastião e Silva, Rend. Mat. Appl., 14, 388 (1955) (Russian transl. Matematika, 1, 60, 1957).
² D. A. Raikov, DAN, 113, No. 5, 984 (1957).
³ A. Grothendieck, Summa Brasil. Mat., 3, 57 (1954).
⁴ N. Bourbaki, Éléments de mathématique, livre V, ch. III–V, Paris, 1955.
⁵ N. Bourbaki, Ann. Inst. Fourier, Grenoble, 2, 5 (1951).
⁶ J. Dieudonné, L. Schwartz, ibid., 1, 61 (1950).
⁷ V. Pták, Czechoslovak Math. J., 3, No. 4, 301 (1954).