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Math Help - Linear Transformations in Normed Spaces

  1. #1
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    Linear Transformations in Normed Spaces

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

    Let V,\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
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  2. #2
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    Quote Originally Posted by Jose27 View Post
    I know this is easy, but I can't find the way to prove this:

    Let V,\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.
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