Let $(X,\langle\cdot,-\rangle)$ be a unitary space, such that for any linear and continuous functional $\varphi:X\rightarrow \mathbb{K}$, there exists $x_0\in X$, such that $\varphi(x)=\langle x,x_0\rangle$ for all $x\in X$. Show that X is a Hilbert space.

My first idea was using, somehow, the Riesz representation theorem, I was looking for converse observations but without any success.

Please help.

  • 1
    $\begingroup$ if the space is not complete, you could complete it to some space $\hat{X}$, then take some $x_0 \in \hat{X}\backslash X$, and consider the map $X \to \Bbb K$ given by $x \mapsto \langle x_0, x \rangle$. That cannot possibly be of the form $\langle x_1,x \rangle$ with $x_1 \in X$. $\endgroup$ – Shalop Feb 19 '17 at 11:24

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