Let $X$ be a $\mathbb{K}$-linear space. $E\subseteq X$. Then $E$ is a basis of $X$ $\iff$ for every $\mathbb{K}$-linear space $Y$ and for every $f:E\rightarrow Y$, there exists a unique $\mathbb{K}$-linear extension $T:X\rightarrow Y$ of $f$.

Is the Hahn-Banach extension theorem is to be used? But I know it for only functional not for maps between spaces.

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    $\begingroup$ What type of basis? And is the extension continuous? $\endgroup$
    – quid
    Sep 16 '16 at 19:43
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    $\begingroup$ If $E$ is a basis then it is easy to construct the extension: whenever $x=\sum_i c_i x_i$ for basis elements $x_i$, $F(x)=\sum_i c_i F(x_i)$. It seems to me that the tricky part is the other direction. $\endgroup$
    – Ian
    Sep 16 '16 at 19:51
  • $\begingroup$ Yes the extension is linear and continuous. $\endgroup$
    – mm-crj
    Sep 16 '16 at 19:55
  • $\begingroup$ What do you mean by $\mathbb{K}$-linear? Do you mean a vector space over $\mathbb{K}$? Or a topological vector space? $\endgroup$
    – Theo
    Sep 17 '16 at 23:40
  • $\begingroup$ @Theo $\mathbb{K}$ is the underlying field of the vector space. $\endgroup$
    – mm-crj
    Sep 19 '16 at 16:59

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