We know If a normed space X is reflexive, then $X'$ is reflexive.and also Reflexive normed spaces are Banach.
but can you proof if X has a finite dimensional Then X is reflexive.
2025-01-13 02:46:07.1736736367
let X has a finite dimensional show that X is reflexive.
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We have $\mathrm{dim}(X) = \mathrm{dim}(X^*) = \mathrm{dim}(X^{**})$. Since the canonical embedding $J : X \to X^{**}$ is an isometry, it is surjective (consider the dimensions). Hence, by definition, $X$ is reflexive.