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Let $H$ be an infinite dimensional separable Hilbert space over $\mathbb{C}$

Let $V \subset H$ be a dense subspace of $H$

Let $f : V \to \mathbb{C}$ be a unbounded functional linear

My question is: Is it always possible extend $f$ to $H$ ? (not necessarily bounded)

Thanks.

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  • $\begingroup$ What do you mean by $f:V\to {\bf C}$ being linear if V is merely a dense subset of H? Are you assuming that V is a dense subspace? $\endgroup$
    – Yemon Choi
    Commented Jan 14, 2022 at 5:26
  • $\begingroup$ @YemonChoi yes $V$ is a dense subspace, sorry now I edit $\endgroup$
    – Matey Math
    Commented Jan 14, 2022 at 6:14
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    $\begingroup$ In that case the answer is yes: take a Hamel basis for V and extend it to a Hamel basis for H. Then define your extension of f by setting it to be zero on all the elements of the Hamel basis that do not lie in V. Of course, this is non-constructive and relies on the axiom of choice $\endgroup$
    – Yemon Choi
    Commented Jan 14, 2022 at 6:19
  • $\begingroup$ OK @YemonChoi your answer is enough for my problem, thank you a lot. $\endgroup$
    – Matey Math
    Commented Jan 14, 2022 at 6:28

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