Finite dimensional normed spaces

[theMR]

Let X and Y be normed spaces. Suppose that $\displaystyle dimX<\infty$. Then every linear operator $\displaystyle T\colon X\rightarrow Y$ is continuous.

How to prove the previous corollary?

 

[theMR]

Sorry, it is corollary to: Any two norms on a finite-dimensional linear space are equivalent.


Definition: Let X and Y be metric spaces. An operator $\displaystyle f:X\rightarrow Y$ is said to be continuous at a point $\displaystyle a\in X$ if, for every $\displaystyle \epsilon >0$, there exists a $\displaystyle \delta >0$ such that:
$$\varrho(x,a)< \delta \Rightarrow \varrho(f(x), f(a))< \epsilon$$