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

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?

## closed as off-topic by Ben McKay, Jan-Christoph Schlage-Puchta, András Bátkai, Ilya Bogdanov, Sean EberhardOct 21 at 10:04

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – Ben McKay, Ilya Bogdanov, Sean Eberhard
If this question can be reworded to fit the rules in the help center, please edit the question.

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.