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Here is mentioned the Riesz Lemma and an important fact on the proof:

Proof of Riesz Lemma

My question is: Everything there is okay for any closed subspace. However, why can't we take $\theta = 1$? Is something related to the dimension of the subspace?

Thank you.

Rub
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    The problem lies in the fact that the unit ball is not compact. In case of finite dimension the unit ball is compact by Heine Borel theorem and hence by elementary analysis extrema are attained. In case of infinite dimension however the unit ball is necessarily not compact and hence u cant conclude that extrema will be attained. – user6 Mar 10 '19 at 09:04
  • But is not the Riesz lemma used to show that the ball is not compact if the dimension is infinite? I mean, it would be like reasoning in circles. Is not there an argument which does not use compactness? Thank you for your answer. – Rub Mar 10 '19 at 09:07
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    Yes it is used but u are asking something stronger than Riesz Lemma. Proving the riesz lemma does not require any condition on the dimension. So u can use the riesz lemma – user6 Mar 10 '19 at 09:09
  • Okay, I think I got it. Thank you very much. – Rub Mar 10 '19 at 09:18
  • I've been thinking and I found a problem. We have that the unit ball is closed and bounded and thus compact and the property holds. However, for a closed subset $Y$, it is not necessarily bounded, isn't it? So how can you ensure that extrema are attained? – Rub Mar 10 '19 at 10:39
  • I try using the basis, but I did not conclude anything. – Rub Mar 10 '19 at 10:39
  • @RubénFernándezFuertes It is not true that being "closed and bounded" implies compactness, that is only true for finite dimensional spaces. Also, in an infinite dimensional vector space you shouldn't think about using basis to prove anything, it usually is not the way to go. – BigbearZzz Mar 10 '19 at 11:06
  • I was asking @SoumilGhosh for the finite-dimensional case because it is the only case for which we can choose $\theta = 1$. But he use only the unit ball and I am asking for a closed subspace in general. – Rub Mar 10 '19 at 11:15

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