My Math Forum Weak Convergence Question

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 January 21st, 2014, 04:30 AM #1 Newbie   Joined: Mar 2013 Posts: 16 Thanks: 0 Weak Convergence Question I just want to confirm that if we have a normed spaces $X$ and $Y$ and $X \subset Y$(X subset Y) then does it follow that if $x_{n} \rightharpoonup x$ weakly in $X$ then $x_{n} \rightharpoonup x$ weakly in $Y$? Thanks.
 January 21st, 2014, 02:16 PM #2 Senior Member   Joined: Dec 2012 Posts: 372 Thanks: 2 Re: Weak Convergence Question $x_n \rightharpoonup x \ \Leftrightarrow \ f(x_n) \rightarrow f(x) \ \forall f \in X^*$. But $X \subset Y \ \Rightarrow \ Y^* \subset X^*$. Given $g \in Y^*$ , then $g \in X^*$ which means $\forall g \in Y^* , g(x_n) \rightarrow g(x)$. Thus, your claim is correct, given that $X$ and $Y$ are both Banach. (Peterpan - have you found some neat application of the weak topology and weak convergence?)

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