Norm-compact unit ball criterion
= Norm-compact unit ball criterion
The closed <unit ball> of a <Hilbert space> is compact in norm exactly when the space is finite-dimensional. Finite-dimensional compactness follows from the <Heine-Borel theorem>. In infinite dimension, an <orthonormal sequence> has pairwise distance $\sqrt2$ and no convergent subsequence. Restricting a <compact operator> that equals a nonzero multiple of the identity on a subspace therefore forces that subspace to be finite-dimensional.