Re: compact are bounded?

On 18 Feb, 16:14, José Carlos Santos <jcsan...@xxxxxxxx> wrote:
On 18-02-2007 14:34, Antonio wrote:

Can you sketch me the following proof, please?

Let E and F be Banach spaces and T : E -> F be a linear map. Suppose T
is compact, i.e., T maps bounded sets into relatively compact sets.
Then T is continuous.

Let B be the closed unit sphere in E. Then T(E) is relatively compact
and therefore is a bounded subset of F.

Is that enough for you?

Best regards,

Jose Carlos Santos

Oh, sure, since compact in a metric space imply bounded.

On the other hand, is it true that a finite rank linear map T : E -> F
(E and F Banach spaces) which is not bounded (i.e., which is not
continuous) is necessarily a compact map?

Thank you, best regards, Antonio.

.

Relevant Pages

  • Civil discourse; Was: question on compact spaces
    ... Every quasicompact space is the surjective image under a continuous map of a compact Hausdorff space. ... I.e. quasicompact just means what most books call compact, though Bourbaki includes the Hausdorff property in the word compact. ... I was intentionally coy with details. ...
    (sci.math)
  • Re: Civil discourse; Was: question on compact spaces
    ... Every quasicompact space is the surjective image under a continuous map of a compact Hausdorff space. ... I.e. quasicompact just means what most books call compact, though Bourbaki includes the Hausdorff property in the word compact. ... whether that was *your* intention or not. ...
    (sci.math)
  • Re: question on compact spaces
    ... Every quasicompact space is the surjective image under a continuous map of a compact Hausdorff space. ... I.e. quasicompact just means what most books call compact, though Bourbaki includes the Hausdorff property in the word compact. ...
    (sci.math)
  • Re: compact are bounded?
    ... is compact, i.e., T maps bounded sets into relatively compact sets. ... is it true that a finite rank linear map T: ...
    (sci.math)
  • Re: compact are bounded?
    ... is compact, i.e., T maps bounded sets into relatively compact sets. ... is it true that a finite rank linear map T: ...
    (sci.math)
Quantcast