Exercise 1069Topology › Metric and normed spacesL1Added 3 Nov 2026

Classical normed spaces

#normed spaces#Banach#examples

Statement

List the classical normed spaces $\ell^p$ and $C([0,1])$ .

Hint

$\ell^p$ sequences with $\sum|x_n|^p<\infty$ ; $C([0,1])$ with $\sup$ -norm.

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$\ell^p$ for $1\le p<\infty$ . $\|x\|_p=(\sum|x_n|^p)^{1/p}$ .

$\ell^\infty$ . $\|x\|_\infty=\sup_n|x_n|$ .

$C([0,1])$ . $\|f\|_\infty=\sup_{[0,1]}|f|$ .

Completeness. All these spaces are Banach (complete normed) spaces.

Conclusion. Classical spaces provide examples and counterexamples in functional analysis.