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If so, how does one go about seeing this?

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In summary, the closed unit ball is not compact in an infinite dimensional normed linear space. This can be proven through various methods, including the Riesz lemma. One idea is to show that the ball is dense in the whole space by proving that the only functional null on the ball is the identically null functional. Another approach is to use the Hilbert space trick, which involves finding a vector that is 'nearly' orthogonal to any proper subspace. Additionally, it can be shown that if the closed unit ball were compact, the normed space would be locally compact and therefore finite dimensional.

- #1

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If so, how does one go about seeing this?

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- #2

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You can prove that the ball is not compact in an infinite dimensional space by invoking the Riesz lemma. (This is one of many ways.)

- #3

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Any other way comes to mind? I wonder if I can prove this with what I know.

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There's a really easy proof for Hilbert spaces (elements of an infinite o.n. set are a distance of sqrt(2) apart).

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(How can elements of an infinite o.n. set be a distance of sqrt(2) apart in the case where the set is a basis and hence dense?)

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And it doesn'T make sense that the ball be dense, since it's closed, it would be the whole space, which it obviously isnt. :p

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Why would this be a problem? If e and f are two distinct elements in any o.n. set, then ||e - f||^2 = <e-f, e-f> = ||e||^2 + ||f||^2 = 2.quasar987 said:(How can elements of an infinite o.n. set be a distance of sqrt(2) apart in the case where the set is a basis and hence dense?)

By the way, the most elementary way to prove this is to use the Riesz lemma. In fact the Riesz lemma basically tells you how to do the Hilbert space trick when you don't have an inner product: it let's you find a vector that's 'nearly' orthogonal to any proper subspace.

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Alright, I'll look into it. There's a nice wiki page about including a proof.

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There is a quite simple intuitive idea behind this, which also leads towards one possible proof. It goes like this:

Let E be a norm space, and B(0,1) the unit ball. Let us assume B(0,1) is compact. Since {B(x,1/2) | ||x|| < 1} is an open cover for the unit ball, there exists a finite number of points x_k so that

[tex]

B(0,1) \subset \bigcup_{k=1}^N B(x_k, \frac{1}{2}).

[/tex]

Now there is two alternatives. Either the x_k span the space E, <x_1,...,x_N>=E, or then they don't. If they do, E is finite dimensional, and we are done. If they do not span, then we end up into a contradiction because the ball B(0,1) has points that have distance greater than 1/2 from the spanned subspace.

You can convince yourself of this by drawing a two dimensional picture. Pretend that x-axis describes the subspace <x_1,...,x_N>, and y-axis some non spanned direction. If you draw a ball x^2+y^2 < 1, you'll see it contains points that have distance greater than 1/2 from the x-axis.

Let E be a norm space, and B(0,1) the unit ball. Let us assume B(0,1) is compact. Since {B(x,1/2) | ||x|| < 1} is an open cover for the unit ball, there exists a finite number of points x_k so that

[tex]

B(0,1) \subset \bigcup_{k=1}^N B(x_k, \frac{1}{2}).

[/tex]

Now there is two alternatives. Either the x_k span the space E, <x_1,...,x_N>=E, or then they don't. If they do, E is finite dimensional, and we are done. If they do not span, then we end up into a contradiction because the ball B(0,1) has points that have distance greater than 1/2 from the spanned subspace.

You can convince yourself of this by drawing a two dimensional picture. Pretend that x-axis describes the subspace <x_1,...,x_N>, and y-axis some non spanned direction. If you draw a ball x^2+y^2 < 1, you'll see it contains points that have distance greater than 1/2 from the x-axis.

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Well it's not in the chapter on normed space (ch. 5). Are you sure you've got the right book?

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look at thm 5.9.4 page 109.

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- #13

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If the closed unit ball were compact, every closed ball would be too, and hence the space would be locally compact, and hence finite dimensional

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