# Linear Transformations in Normed Spaces

• Jun 3rd 2009, 07:05 PM
Jose27
Linear Transformations in Normed Spaces
I know this is easy, but I can't find the way to prove this:

Let $\displaystyle l:(V,\Vert . \Vert _V) \longrightarrow (W,\Vert . \Vert _W)$ be a linear transformation. Show that if $\displaystyle l$ is continous at $\displaystyle 0$ then there exists $\displaystyle c>0$ such that $\displaystyle \Vert l(v) \Vert _W \leq c \Vert v \Vert _V \forall v \in V$
• Jun 4th 2009, 08:44 AM
NonCommAlg
Quote:

Originally Posted by Jose27
I know this is easy, but I can't find the way to prove this:

Let $\displaystyle l:(V,\Vert . \Vert _V) \longrightarrow (W,\Vert . \Vert _W)$ be a linear transformation. Show that if $\displaystyle l$ is continous at $\displaystyle 0$ then there exists $\displaystyle c>0$ such that $\displaystyle \Vert l(v) \Vert _W \leq c \Vert v \Vert _V \forall v \in V$

obviously we may assume that $\displaystyle v \neq 0.$ since $\displaystyle \ell$ is continuous at 0, there exists $\displaystyle \delta >0$ such that if $\displaystyle ||x||_V < \delta,$ then $\displaystyle ||\ell(x)||_W < 1.$ now choose $\displaystyle c =\frac{2}{\delta}$ and $\displaystyle x=\frac{v}{c||v||}, \ \ 0 \neq v \in V.$