# Universal Property of Basis

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.

• What type of basis? And is the extension continuous?
– quid
Sep 16 '16 at 19:43
• 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. 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. Sep 19 '16 at 16:59