Consider the metric space $(C[0,1],||.||_{\infty})$ , in this space , is the set
$\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded ?
Please help , Thanks in advance
2026-04-21 04:39:34.1776746374
In $(C[0,1],||.||_{\infty})$ , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded?
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It's not even bounded. Consider the functions $(n+1)x^n, n = 1,2,\dots $