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

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?

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.