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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.

Mini_me
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mmcrjx
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    What type of basis? And is the extension continuous? – quid Sep 16 '16 at 19:43
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    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. – Ian Sep 16 '16 at 19:51
  • Yes the extension is linear and continuous. – mmcrjx Sep 16 '16 at 19:55
  • What do you mean by $\mathbb{K}$-linear? Do you mean a vector space over $\mathbb{K}$? Or a topological vector space? – Theo Sep 17 '16 at 23:40
  • @Theo $\mathbb{K}$ is the underlying field of the vector space. – mmcrjx Sep 19 '16 at 16:59
  • This is true for vector spaces in general. It's essentially the universal property of free spaces. I'm not sure how this would work for topological vector spaces. – Ethan Dlugie Oct 06 '20 at 20:27

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