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I was trying to prove the following statement

"A closed subspace $Y$ of a reflexive Banach space $X$ is also reflexive."

I followed the steps similar to this answer on math.stackexchange

  • Since the canonical map from $J_X: X \to X^{**}$ (sending $x \in X$ to $g_x \in X^{**}$ defined by $g_x(f)=f(x)$) is an isometry and an injection, it suffices to show that $J_Y: Y \to Y^{**}$ is surjective
  • Let $\eta \in Y^{**}$, we define a linear functional $\xi \in X^{**}$, such that $\xi(f)=\eta(f|_Y), \forall f \in X^*$
  • It follows from the reflexivity of $X$ that $\xi(f) = g_x(f) = f(x)$ for some $x \in X$
  • We show that $x\in Y$ by considering the annihilator of $Y$ as suggested in the post linked above
  • Now we see $\eta(f|_Y)=g_y(f|_Y)$ for every $f \in X^*$

My questions are:

  • How is it possible to show that every linear functional on $Y$ is a restriction of some functional on $X$? The post linked above seems to suggest using the Hahn-Banach theorem (so do some other sources on the internet) but I fail to see how this can be applied, as the Hahn-Banach theorems require the functional to be bounded by some semi-norm.

  • Why is $X$ being a Banach space required here?

Maybe I am missing something trivial here and this should really be a comment under that post linked above instead of a separate question but I do not have enough reputation to comment.

isomorphicdude
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1 Answers1

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About the first question: you are missing one fundamental aspect, the fact that we are considering just continouous linear functionals on $Y$, which verify the bound $$|f(y)| \leq ||f|| \mbox{ }||y||$$ and therefore can be extended to continuous linear functionals on $X$ keeping this bound (and so keeping the norm).

About the second question: you are right, the Banach hypotesis is not required. Actually every closed subspace of a reflexive normed space is reflexive.

Dark Magician
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    The fact is that in functional analysis one often restricts their attention to Banach spaces since there the Baire category theorem and its powerful consequences (e.g. the Banach-Steinhaus theorem, the open mapping theorem, the closed graph theorem) are available. But some results are true also for normed spaces. – Dark Magician Sep 01 '22 at 08:12