Re: compact are bounded?

From: José Carlos Santos <jcsantos@xxxxxxxx>
Date: Sun, 18 Feb 2007 15:14:40 +0000

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
.

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