Hilbert spaces

• Why an Infinite-Rank Compact Operator Has Nonclosed Range
  A compact operator between Banach spaces with closed range must have finite rank. The quotient-space proof explains why infinite rank forces nonclosed range.
• What Does the Polarization Identity Imply for Operators?
  The polarization identity shows that a linear norm-preserving map on a Hilbert space preserves inner products and satisfies T* T = I.