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Let $X,Y$ be normed linear spaces , $W$ be a linear subspace of $X$ , let $T:W \to Y$ be a continuous linear tranformation with finite rank i.e. $T(W)$ is finite dimensional ; then can we extend $T$ to a continuous linear transformation $\bar T :X \to Y$ such that $\bar T(X)=T(W)$ ?

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Yes, let $P: Y\to T(W) $ be a continuous projection on $T(W),$ and let $A: X\to Y $ be extension of $T$ then define $\tilde{T} : X\to Y,$ $\tilde{T} = P\circ A .$

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    How do you get a continuous extension of $T$ ? And though $T(W)$ is finite dimensional , $Y$ is not Banach , so how do you get a continuous projection ? –  Apr 17 '16 at 07:10
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    Since, $T(W)$ is finite dimensional, then there exist a continuous linear functional $f_1 , f_2, ... , f_n : W\to \mathbb{R}$ such that $T(w) =\sum_{j=1}^n f_j (w) e_j $ for some linearly independent vectors $e_j \in Y.$ Now using Hahn - Banach theorem we can extend continously each $f_j $ to a $\tilde{f}j : X\to\mathbb{R}$ and the operator $\tilde{T} : X\to T(W) , T(x)=\sum{j=1}^n \tilde{f}_j (x) e_j $ is a continuous extension of $T$ –  Apr 17 '16 at 11:22
  • For continuous projections look here:http://math.stackexchange.com/questions/156609/continuous-projections-to-finite-dimensional-subspaces-of-normed-spaces –  Apr 17 '16 at 11:24
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    How do we know that each $f_j$ is continuous ? –  Apr 18 '16 at 04:05
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    It folows from the fact that on finite dimensional subspace $T(W)$ all norms are equivalent for example there are equivalent to the norm $$||\sum_{j=1}^n x_i e_i || =\sup_{1\leq i\leq n } |x_i| $$ –  Apr 18 '16 at 05:05