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Let $H$ be a separable Hilbert space (of non-zero dimension), let $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $L^2(\mu;H)$ be the Bochner-space $\mu$-integrable $H$-valued functions. ...

Let $X$ be a Banach space. By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball. I am ...