Coincidence of the Weak/Norm Topologies on a Normed Linear Space

# Coincidence of the Weak and Norm Topologies on a Normed Linear Space

Recall that if $X$ is a normed linear space then the weak topology on $X$ is the weakest topology which makes every $f \in X^*$ continuous with respect to the topology. We noted that the weak topology on $X$ is weaker than the norm (strong) topology on $X$.

That said, is it possible for the weak topology on $X$ to coincide with the norm topology on $X$, and if so, when? The following theorem answers that question.

Theorem 1: Let $X$ be a normed linear space. Then the weak topology on $X$ is equal to the norm topology on $X$ if and only if $X$ is finite-dimensional. |