# Thread: Linear Transformations in Normed Spaces

1. ## Linear Transformations in Normed Spaces

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

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

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

Let $lV,\Vert . \Vert _V) \longrightarrow (W,\Vert . \Vert _W)" alt="lV,\Vert . \Vert _V) \longrightarrow (W,\Vert . \Vert _W)" /> be a linear transformation. Show that if $l$ is continous at $0$ then there exists $c>0$ such that $\Vert l(v) \Vert _W \leq c \Vert v \Vert _V \forall v \in V$
obviously we may assume that $v \neq 0.$ since $\ell$ is continuous at 0, there exists $\delta >0$ such that if $||x||_V < \delta,$ then $||\ell(x)||_W < 1.$ now choose $c =\frac{2}{\delta}$ and $x=\frac{v}{c||v||}, \ \ 0 \neq v \in V.$