= Completely continuous operator
{title2=$x_n\rightharpoonup0\Longrightarrow\|Tx_n\|\to0$}
A <bounded linear operator> between <Banach spaces> is completely continuous when it maps weakly convergent sequences to norm-convergent sequences. Equivalently, it maps weakly compact sets to norm-compact sets: the <Eberlein-Šmulian theorem> gives the forward direction, while a weakly convergent sequence together with its limit is weakly compact and gives the reverse direction. The inclusion $C(\mathbb T)\to L^2(\mathbb T)$ for finite <Lebesgue measure> is completely continuous by pointwise convergence, the <Uniform boundedness principle> and the <dominated convergence theorem>, but is not a <compact operator> because of the orthogonal Fourier modes.
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