Let X be a Banach space which has no infinite reflexive subspaces.
Does this assumption implies that the space X itself is not reflexive?
See, e.g., Nelson Dunford, Jacob T. Schwartz, Linear Operators I, II.3.23:
23 THEOREM. A closed linear manifold in a reflexive B-space is
reflexive.
or Walter Rudin, Functional Analysis - McGraw-Hill (1991), p. 111, Exercises, 1. (d) ...every norm-closed subspace of a reflexive space X is
reflexive.
See, e.g., An Introduction to Banach Space Theory, Robert E. Megginson. Pag. 99-109, 410-411, 441.
Let X be a Banach space which has no infinite dimensional reflexive subspaces. Then either X is finite dimensional or else it must be non-reflexive.
If X is an infinte dimensional reflexive space, then every closed subspace of it is also reflexive. Of course every separated locally convex (then also normed) space has closed subspaces (kernels of continuous functionals for example). Therefore the answer is yes. Your space cannot be reflexive.
Let Y be a null space of some non-zero continuous linear functional on X. Then Y is a proper closed subspace of X. From the assumption, it follows that Y is not reflexive.
Suppose that X is reflexive. Then, it is easy to see that Y is also reflexive. Hence X is not a reflexive space.
Something is missing in the statement of the question. In the way how it is formulated now
"Let X be a Banach space which has no infinite(-dimensional) reflexive subspaces. Does this assumption imply that the space X itself is not reflexive?"
the answer is trivial: If X is finite-dimensional, then it has no infinite-dimensional reflexive subspaces, but X itself is reflexive.
If one reformulates the question adding that X itself is infinite-dimensional, then X also is a subspace of X, so by the assumption of the question, X cannot be reflexive.
Finally, if one asks the question "Let X be an infinite-dimensional Banach space which has no PROPER infinite-dimensional reflexive subspaces. Does this assumption imply that the space X itself is not reflexive",
then the answer really needs the theorem that every closed subspace of a reflexive Banach space is reflexive, which can be found in most functional analysis textbooks, for example in those which were recommended in the answer by Oleg Reinov.
Indeed, the original question was formulated in a sloppy way. Almost any person who was thinking about this problem for a moment tacitly assumed the setup Vladimir just indicated. This is why we need to make sure that a formulation is exact and formally correct.
Thank you, Vladimir, for putting it straight.
It is true that the original question was posed in a somewhat sloppy way.
The revised question posed by Vladimir whose answer comes quickly using
the well known result
" A closed subspace of a reflexive Banach space is reflexive"
is a much better alternative.
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