Linear transformation in space of infinite dimension

Question

Problem.

(a) Show that every linear transformation $A: \mathbb{R}^{n} \to \mathbb{R}^{m}$ is continuous.

(b) If we change $\mathbb{R}^{n}$ by a normed space $X$ and take $m=1$, the item (a) is true?

Solution. For item (a), is easy to show that $A$ is Lipschitz. For (b), is $\dim X$ is finite, take an isomorphism $T: \mathbb{R}^{p} \to X$ where $\dim X = p$ and $T(e_{i}) = u_{i}$ ($\{e_{i}\},\{u_{i}\}$ are bases for $\mathbb{R}^{p},X$ respectively). My question is when $\dim X = \infty$. There is spaces of infinite dimension such that a linear transformation may not be continuous, but in this specific case, I don't know. Can someone help me?

Answer 1

A classical example consists in taking:

• $X=C\bigl([0,1]\bigr)$;
• if $f\in X$, $\|f\|=\int_0^1|f|$;
• $\psi(f)=f(1)$.

Then $\psi$ is discontinuous: if $f_n(t)=t^n$, then $\lim_{n\to\infty}f_n=0$, but $\lim_{n\to\infty}\psi(f_n)=1\neq\psi(0)$.