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I know that there is a lemma which tells me that if $V$ is a closed subspace of a Hilbert space $H$, if $y \in H$, and $y \notin V$, then if $$V^* = \text{Linear span}\ (V, y)$$we have $V^*$ is a closed subspace of $H$.

Now i have a confusion that in the proof of above lemma nowhere we talk about the dimension of $V$, so it might holds for infinite dimension also.

Let $H = L^2([0,1])$ and let $P$ be the set of polynomials in $H$. Since i know the basis of $P$ which are $\{1, x, x^2, \ldots\}$. By above lemma if we take $V = \{0\}$, then by continuously applying lemma we have $P$ must be closed in $H$ which is not in actual. So where i am wrong in approaching this?

kapil
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1 Answers1

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You can apply the lemma $n$ times to prove that $\{1,x,x^2,...,x^{n-1}\}$ is closed, but it doesn't hold at infinity. It would be like saying that the union of infinitely many closed sets is closed. In fact $P$ is the union wity $n$ from $0$ to infinity of $\{1,...,x^{n}\}$

Emilio
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  • But Why can't we apply it countable times? – kapil Oct 13 '16 at 12:44
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    The lemma can be applied as many times as you want, but the dimension of the resulting subspace will be finite. As big as you want, but that doesn't imply infinite dimension. For an easier example, think of a finite set and the fact that after adding one element it is still finite. You can add an element as many times as you want and the result is a finite set. That doesn't imply that you can add a countable infinit amount of elements and still get a finite set – Emilio Oct 13 '16 at 13:41