Reflexivity of $(C^1([a,b]),\left\lVert \cdot\right\rVert_{\infty})$

A non-reflexive Banach space $(X,\|\cdot\|)$ can't have a weaker norm $|\cdot |$ making it reflexive, because reflexive normed spaces are complete (the dual of any normed space is complete for the dual norm) and thus, Banach's isomorphy theorem (the open mapping theorem) implies that both norms $\|\cdot\|$ and $|\cdot|$ are equivalent so that $(X,\|\cdot\|)$ would be reflexive.

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General Topology

Functional Analysis

Reflexive Space

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