I've just proved the fact that every linear function on a finite dimensional normed vector space is uniformly continuous. Because, let $T:U\to V$ be a linear function on $U$ with basis : $(u_1,u_2,\dots,u_n)$ and suppose that $\|\cdot\|_u$, $\|\cdot\|_v$ be the respective norms associated with $U$ and $V$. Setting $M= \max(\|Tu_1\|_v,\dots,\|Tu_n\|_v)$, we have $$\|Tu\|_v=\|a_1Tu_1 + a_2Tu_2 +\dots+ a_nTu_n\|_v \le |a_1|\cdot\|Tu_1\|_v+\dots+|a_n|\cdot\|Tu_n\|_v \le M\|u\|_1,$$ where $\|u\|_1=|a_1|+\dots+|a_n|$. Since all norms on $U$ are equivalent, we have $\|Tu\|_v\le C\|u\|_u$ for some $C>0$. Thus, $\|Tx-Ty\| \le C\|x-y\|$ for all $x$, $y$ in $U$, thereby satisfying Lipschitz' condition.
My questions are:
- Is it true that a linear function from one vector space to another is always continuous? (finite dimension not assumed)
- If so, or else, is a continuous linear map always uniformly continuous?