Re: closed bounded in Banach space


José Carlos Santos ha scritto:

Sauro wrote:

Is a closed bounded subset of a Banach space also compact?
No. Take the closed unit ball in any infinite-dimensional
Banach space. It is closed and bounded, but not compact.

Is a closed subset S of a Banach space also compact (I assume such a S
is a subset of a finite dimensional subspace of our Banach space)?

If your question is "is every closed subset of every finite-demensional
subspace of a Banach space compact?" then the answer is "yes".

Best regards,

Jose Carlos Santos

Excuse me Carlos, but, thinking more closely to my question "is every
closed subset of every finite-demensional subspace of a Banach space
compact?", it doesen't seem to be true, consider for example the set
{(x,y) in |R^2 such that x lies in [0,1] and y lies in [0, + infty ) }.
Then such a set is a closed subset of |R^2, which itself is a
finite-demensional subspace of a Banach space, |R^2. On the other hand
{(x,y) in |R^2 such that x lies in [0,1] and y lies in [0, + infty ) }
is cleary not compact...

What do you think about?

Regards, S.

.

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