= Compact operators send weak convergence to norm convergence
{title2=$f_n\rightharpoonup f\Longrightarrow Kf_n\to Kf$}
For a bounded operator $K$ between <Hilbert spaces>, compactness is equivalent to $f_n\rightharpoonup f$ implying $Kf_n\to Kf$ in norm. The <Uniform boundedness principle> bounds a weakly convergent sequence. Relative compactness of its images and the unique possible weak limit then give norm convergence. Conversely, every bounded domain sequence has a weakly convergent subsequence, so the image of the <unit ball> is relatively compact.
Back to article page
