I am trying to show from first principles that a linear map $T:X \rightarrow Y$ between finite dimensional vector spaces $X$ and $Y$ is Lipschitz. I'm rather stuck and would appreciate a hint how I might continue the argument I have constructed so far:
To show that $T$ is Lipschitz, it is necessary to show for $x,y \in U$ that $|T(x) - T(y)| \leq \lambda|x - y|$ for some $\lambda > 0$. If $u_1, \dots, u_n$ is a basis for $X$, there exists scalars $a^i, b^i$ such that $x = a^iu_i$ and $y=b^iu_i$. Therefore,
$ |T(x - y)| \leq |T(a^iu_i)| + |T(b^iu_i)| = |a^i|\cdot|T(u_i)| + |b^i|\cdot|T(u_i)| \leq k \sum_{i=1}^n|T(u_i)| $
where
$k = \max_{j=1, \dots n}|a^j + b^j|$
So, I've succeeded in finding a bounds of sorts for $T$ but I'm not sure how to translate the bound $|T(u_i)|$ to $u_i$ and other than showing something rather obvious so far, I'm not really sure that the above argument gets me a lot closer to my goal. How could I proceed with this?