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?



Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence?
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.


