Questions are from the theory of PDEs\SPDEs
Question 1. Suppose $(V, H, V^\star)$ is a Gelfand triple (embeddings are continuous and dense, so $\|\|_H \le C \|\|_V$ for some $C>0$ etc) of separable Hilbert spaces. Consider a probability space $(\Omega, F, P)$ and define the following spaces (of measurable mapping) for $p \ge 1$ $$ L_p(X) = \{f: [0;T] \times \Omega \mapsto X \mid E \int_{[0;T]} \| f(t)\|_X^p dt < \infty\}, \quad X \in \{V, V^\star, H\}. $$
Then $$ L_p(X)^\star = L_q(X^\star) $$ where $q = \frac{p}{p-1}, p > 1.$
Suppose $(u_n)_n$ is bounded in $L_2(H)$ and $L_p(V).$ Then (by the Banach-Alaoglu theorem) there exists a subsequence $(u_m)$ s.t. $u_m \to u$ in $L_2(H)$ weakly and $u_m \to v$ in $L_p(V)$ weakly.
Why $u=v$ a.e.?
Question 2. Suppose $(u_n)$ is bounded in $L_2(\Omega, C([0;T], H)).$ Why is it true that there is a subsequence $u_m$ converging weak* in $L_2(\Omega, L_\infty([0;T], H))?$ What is the dual for the first space?
I would strongly appreciate any references that provide a good overview of properties of such spaces.
Thank you.
