Generalization of Hahn-Banach theorem: given $f : U \to W$, can we always extend it to get $g : V \to W$? We know Hahn-Banach theorem:
"If $V$ is a normed $K$-vector space with linear subspace $U$ (not necessarily closed) and if $f : U \to K$ is continuous and linear, then there exists an extension $g : V \to K$ of $f$ which is also continuous and linear and which has the same norm as $f$."

Does it stay true if we replace $K$ by any $K$-vector space $W$? Or is there a counter-example? To be clear: is there any example of a normed space $V$, a subspace $U \subset V$, a normed space $W$ and a continuous linear map $f : U \to W$ which has no extension $V \to W$?

(I'm trying to see if, given any normed vector spaces $V$ and $W$, it is possible to build a non-trivial continuous linear map $g : V \to W$. Is it always possible? Apparently there are sufficiently many such $g$, see here (and using Hahn-Banach theorem to get $X' \neq \{0\}$)).
Thank you very much!
 A: Hint: Consider $(C^0([0,1],\| \|_{\infty})$ the space of continuous functions on $[0,1]$ and $P([0,1])$ the space of polynomial functions on $[0,1]$. Consider $d:P([0,1]\rightarrow C([0,1])$ the derivative, it is continue. But you cannot extend it to $C([0,1])$ 
since Stone Weirstrass implies that a continue function is  a limit of polynomial. Take the continue $f$ such that $f(x)=2x, x<1/2, f(x)=x+1/2, x\geq 1/2$, it is continue and it is the limit of the sequence of polynomials $p_n$ by Stone Weirstrass, this implies that $lim {p_n}_{\mid [0,1/2]}=2x$ and $lim_n{p_n}_{\mid [1/2,1]}=x+1/2$, if $df$ is defined, $df_{\mid[0,1/2]}=lim_nd{p_n}_{\mid [0,1/2]}=2, df_{\mid[1/2,1]}=1$ impossible since this function is not continue. 
A: Whenever a Banach space $V$ (e.g. $V=\ell^\infty$) has a non-complemented subspace $U$ (e.g. $U=c_0$) then the identity $U \to U$ does not have a continuous linear extension $V\to U$ just because this would be a projection onto $U$ (which does not exist by definition of complemented subspaces).
