How to find if it is a compact operator:
$F\colon C[0,1]\rightarrow C[0,1]$ : $x(t)\mapsto \int^1_0 \cos(t^2+s^2)x(s)ds$
Could you please help with this question.
2026-03-26 07:32:30.1774510350
How to find if it is a compact operator
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Notice that $$F(x)(t)=\cos(t^2)\int_0^1\cos(s^2)x(s)\mathrm ds-\sin(t^2)\int_0^¹\sin(s^2)x(s)\mathrm ds,$$ hence the range of $F$ is contained in the finite dimensional subspace generated by the functions $t\mapsto \cos(t^2)$ and $t\mapsto \sin(t^2)$.
This implies that $F$ is compact.