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In a theorem I am reading about closed subspace the author states that an infinite dimensional subspace need not be closed.

What is an example of infinite dimensional subspace that is not closed?


marked as duplicate by Carsten S, E. Joseph, Dominik, Alex M., user228113 Dec 23 '16 at 11:11

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  • $\begingroup$ Perhaps better stated as "In a a normed linear space $X$ an infinite dimensional subspace need not be closed in $X.$ $\endgroup$ – zhw. Dec 23 '16 at 0:34

Take $C([0,1],\|\|_{\infty})$ and the subset of polynomials. Every continuous function is a limit of polynomials by Stone Weirstrass. Thus the subset of polynomial functions of $C([0,1])$ is dense, thus it is not closed.

  • 2
    $\begingroup$ nit picky: "dense and a proper subset, thus not closed". The whole space is closed and dense $\endgroup$ – user2520938 Dec 23 '16 at 9:42

Let $\ell^2$ be the space of all square-summable real (or complex) sequences $x = (x_1,x_2, \ldots)$ with norm $\|x\| = \displaystyle ( \sum |x_i|^2)^{1/2}$. Let $V \subset \ell ^2$ be the subspace of all sequences with all but finitely many entries equal to zero. Then $V$ is infinite-dimensional but not closed. It is not closed because its closure contains the limit point $(1,1/2, 1/3, \ldots)$


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