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Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent subsequence, say $f_{n_k}$, converging to $f$. Can we say that $f_n\to f$ strongly?

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Let $(f_n)$ be any sequence which converges to $f$ weakly but not strongly. Define a new sequence $(g_k)$ by setting $g_{2n} = f_n$ and $g_{2n+1} = f$. Then $(g_k)$ shows that the answer to the question is negative.

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