Dilatation (?!)

[Principe2]

Let $B$ a Banach space such that $|B|\le|RR|$. Say if there exists a function $f : B->RR$ such that

$||f(x)-f(y)||_{R}\ge||x-y||_B, \forall x,y\inB$

[Luca.Lussardi]

In the codes theory there are some interesting example: I will search something....

[Chevtchenko]

"Luca.Lussardi":The general definition of normed space does not require the real or complex fileld, but a general field $K$ with a suitable definition of absolute value to make sense the definition of the norm.

I didn't know that! Could you please write out this definition?

[Luca.Lussardi]

The general definition of normed space does not require the real or complex fileld, but a general field $K$ with a suitable definition of absolute value to make sense the definition of the norm.

[Chevtchenko]

A little remark. If $V$ is a vector space over some field $k$, and $V \ne {0}$, then $|V| >= |k|$. To see this, take any $x \in V - {0}$; then plainly the map $\lambda \in k \mapsto \lambda x \in V$ is injective.

Now if I recall correctly a normed space (or for that matter a Banach space) is supposed to be a vector space over the real or the complex field... hence it cannot be finite (except in the trivial case).

[Luca.Lussardi]

You can consider a finte vector space (in the Galois theory there are some examples) with a norm.

[Chevtchenko]

"Luca.Lussardi":Let $B$ be a finite normed space; then $B$ is complete since every Cauchy sequence $x_n$ in $B$ is constant for $n$ sufficiently large.

Will you be so kind as to exhibit an example of a finite normed space (apart from ${0}$)?

[Luca.Lussardi]

Let $B$ be a finite normed space; then $B$ is complete since every Cauchy sequence $x_n$ in $B$ is constant for $n$ sufficiently large.

[Chevtchenko]

"Luca.Lussardi":Let $B$ be a finite subset of $\RR$: is it trivial to find $f$?

Unless $B = {0}$, how could $B$ be finite and a Banach space?

[Principe2]

in this case, it is sufficient to choose $f=id$.

Ops. Avevo scritto le norme al contrario. Ho corretto!

[Luca.Lussardi]

Let $B$ be a finite subset of $\RR$: is it trivial to find $f$?