Show that $T$ is a compact operator

real-analysis

Solution 1:

Hints:

  1. $TB = \{ g\in C^1[0,1] \mid \|g'\| < 1, g(0) = 0 \}$
  2. $TB$ is uniformly Lipschitz continuous by the Lipschitz constant $1$, in particular it is equicontinuous.
  3. $TB$ is uniformly bounded, as for $g\in TB$ we have $$ |g(x)| \le \int_0^x |g'(t)| dt \le x \le 1. $$
  4. By Arzela Ascoli theorem $TB$ is pre compact.
  5. For $g_k \in \overline{TB}$ there exists a $h_k\in TB$ with $\|g_k-h_k\| < 1/k$. Then, $h_k$ has a convergent subsequence and the corresponding subsequence of $g_k$ has the same limit.

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