Are closed subspaces of reflexive locally convex spaces reflexive?

Question

We know that if $X$ is a Banach space which is reflexive, then any closed subspace of $X$ is reflexive, could we extend the conclusion to any locally convex topological vector space, where $X^{'},X^{''}$ equipped with strong topology?

Answer 1

No, closed subspaces of a reflexive locally convex linear space are not necessarily reflexive. This is already mentioned in Dieudonné's paper Recent developments in the theory of locally convex vector spaces from 1953.

Theorem 20.6 of Kelley's book Linear Topological Spaces characterises reflexive spaces as follows:

A locally convex Hausdorff space is reflexive if and only if it is semi-reflexive (bounded weakly closed sets are weakly compact) and evaluable (strongly bounded subsets of the adjoint are equicontinuous).

One can proof, e.g., Theorem 20.2 in Kelley's book, that closed subspaces of semi-reflexive spaces are again semi-reflexive. However, a similar statement does not hold for evaluable spaces.

A counterexample showing that a closed subspace of a reflexive locally convex space is not necessarily reflexive can be found as Exercise 20.D of Kelley's aforementioned book.

Answer 2

Unfortunately, a reflexive locally convex space may have a non-reflexive closed subspace. This is Corollary 27.24 in D. Vogt and M. S. Ramanujan's Introduction to functional analysis.

However, if you assume additionally that your space is Fréchet, then closed subspaces will be automatically reflexive.