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Let $H$ be a Hilbert space. Prove that the following are equivalent:

a) the algebraic dimension of $H$ is finite;

b) each closed, not empty subset $C$ has an element of minimum norm (that is the solution of $\inf _{x\in C} || x||$).

I have managed to prove that a)$\implies$b) but I don't have any idea for the other implication. I would appreciate it if anyone could help me. Thank you.

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If the algebraic dimension is infinite, there exists an orthonormal sequence $(e_n)_{n\geqslant 1}$. Define $$C:=\left\{\frac{n+1}ne_n,n\geqslant 1\right\}.$$ Then $C$ is closed and an element of minimal norm should be of norm $1$.

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  • $\begingroup$ @ Davide Giraudo: Thank you, but why is C closed? $\endgroup$ – user62138 Jan 13 '14 at 16:20
  • $\begingroup$ A convergent sequence of elements of $C$ is necessarily stationary because $\min\{\lVert x-y\rVert, x,y\in C,x\neq y\}\geqslant 1$. $\endgroup$ – Davide Giraudo Jan 13 '14 at 16:42

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