I am considering the normed linear space $C[0,1]$ with norm $$\lVert x\rVert_{\infty} = \max\{\vert{x(t)}\vert: t\in [0,1]\}.$$

Now I want to show that if the sequence $\{x_n\}_{n\in \mathbb{N}}$ in $(C[0,1], \lVert\cdot\rVert_{\infty})$ converges to $x \in C[0,1]$, then it also converges pointwise to $x$.

Ideally I would like to understand how the solution is formed, I have seen other examples by my lecturer but these were specific ones and I am struggling to apply it to this general case. Any help is greatly appreciated!

  • 2
    $\begingroup$ You have $|x(t)| \le \|x\|_\infty$ so pointwise convergence is immediate. In particular, $|x(t)-x_n(t)| \le \|x-x_n\|_\infty$. $\endgroup$ – copper.hat Feb 7 '18 at 18:44

Yes, it is true. Take $\varepsilon>0$. Now, take $p\in\mathbb N$ such that$$n\geqslant p\implies\|x-x_n\|<\varepsilon.$$Then, for each $t\in[0,1]$,$$n\geqslant p\implies\bigl|x(t)-x_n(t)\bigr|\leqslant\|x-x_n\|<\varepsilon.$$


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