I've a problem with some exercise, namely:

Show that if X is a finite-dimensional Banach space, then every linear functional f on X is continuous on X.

Use Proposition: Every operator T from a finite-dimensional normed space X into a normed space Y is continuous.

I don't even know how to start...

Can someone help?
Thanks and regards!


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    $\begingroup$ A linear functional is a linear operator. With codomain the scalar field. So this is a particular case of the proposition you quote. $\endgroup$ – Julien May 10 '13 at 14:04

Recall that all norms are equivalent in finite dimensional vector space and let $(e_1,\cdots,e_n)$ a basis for $X$ and let $$x=\sum_{i=1}^n x_i e_i\in X$$ then we have $$|f(x)|=\left|\sum_{i=1}^n x_if( e_i)\right|\leq \sum_{i=1}^n |x_i| |f( e_i)|\leq M \sum_{i=1}^n |x_i|=M||x||_1$$ where $$M=\max_{1\le i\le n}|f(e_i)|$$

and then we can deduce.

  • $\begingroup$ Thanks! You really helped me. The rest also thanks $\endgroup$ – Dareq May 11 '13 at 14:29
  • $\begingroup$ You're welcome. $\endgroup$ – user63181 May 11 '13 at 16:31

Hint: A linear functional is a linear operator into the scalar field.


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