# Continuity and norm space

• 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 $\displaystyle$\ I_d :\left( {V,\left\| x \right\|_1 } \right) \to \left( {V,\left\| x \right\|_2 } \right)\$$es continuos iff the set A\displaystyle \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 $\displaystyle$\ I_d :\left( {V,\left\| x \right\|_1 } \right) \to \left( {V,\left\| x \right\|_2 } \right)\$$es continuos iff the set A\displaystyle \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, $\displaystyle \text{id}:V\to V$ will be bounded if and only if there is a constant $\displaystyle M$ such that $\displaystyle \|\text{id}(v)\|_1\leqslant M\|v\|_2$ for all $\displaystyle v\in V$... now why is this equivalent to saying that this inequality holds true for every $\displaystyle v\in B_{\|\cdot\|_1}(0;1)$?
• 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, $\displaystyle \text{id}:V\to V$ will be bounded if and only if there is a constant $\displaystyle M$ such that $\displaystyle \|\text{id}(v)\|_1\leqslant M\|v\|_2$ for all $\displaystyle v\in V$... now why is this equivalent to saying that this inequality holds true for every $\displaystyle v\in B_{\|\cdot\|_1}(0;1)$?

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

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

Originally Posted by Drexel28
Ok, so then what have you been taught?

Here is what Ive been taught.
• May 25th 2011, 02:33 PM
Drexel28
Quote:

Originally Posted by orbit
Here is what Ive been taught.

I don't mean to be rude but