Continuity and norm space

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May 23rd 2011, 07:52 PM
orbit

Continuity and norm space

Let V a norma space with 2 norms , lets say norm 1 and norm 2

Show that the function identity $\ I_d :\left( {V,\left\| x \right\|_1 } \right) \to \left( {V,\left\| x \right\|_2 } \right)\$ es continuos iff the set
A $\A = \left\{ {x \in V/\left\| x \right\|_1 = 1} \right\}\$$ is bounded with norm 2

Please somebody give me a biiiiiiiiiiiiig hint....

Regards...
May 23rd 2011, 09:18 PM
Drexel28

Quote:

Originally Posted by orbit

Let V a norma space with 2 norms , lets say norm 1 and norm 2

Show that the function identity $\ I_d :\left( {V,\left\| x \right\|_1 } \right) \to \left( {V,\left\| x \right\|_2 } \right)\$ es continuos iff the set
A $\A = \left\{ {x \in V/\left\| x \right\|_1 = 1} \right\}\$$ is bounded with norm 2

Please somebody give me a biiiiiiiiiiiiig hint....

Regards...

Come on man--you really need to show some effort. You need remember that linear operators are continuous if and only if they're bounded. So, $\text{id}:V\to V$ will be bounded if and only if there is a constant $M$ such that $\|\text{id}(v)\|_1\leqslant M\|v\|_2$ for all $v\in V$ ... now why is this equivalent to saying that this inequality holds true for every $v\in B_{\|\cdot\|_1}(0;1)$ ?
May 24th 2011, 04:36 PM
orbit

Quote:

Originally Posted by Drexel28

Come on man--you really need to show some effort. You need remember that linear operators are continuous if and only if they're bounded. So, $\text{id}:V\to V$ will be bounded if and only if there is a constant $M$ such that $\|\text{id}(v)\|_1\leqslant M\|v\|_2$ for all $v\in V$ ... now why is this equivalent to saying that this inequality holds true for every $v\in B_{\|\cdot\|_1}(0;1)$ ?

Hi, thanl you for answering. unfortunately for me,linear operators haven´t been taught is my course.
May 24th 2011, 09:35 PM
Drexel28

Quote:

Originally Posted by orbit

Hi, thanl you for answering. unfortunately for me,linear operators haven´t been taught is my course.

Ok, so then what have you been taught?
May 25th 2011, 01:16 PM
orbit

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Quote:

Originally Posted by Drexel28

Ok, so then what have you been taught?

Here is what I`ve been taught.
May 25th 2011, 02:33 PM
Drexel28

Quote:

Originally Posted by orbit

Here is what I`ve been taught.

I don't mean to be rude but

A) I'm not going to read a whole book to help you with one problem.

B) That's in Spanish man.