If M and N are orthogonal subspaces of a Hilbert space, then M + N is closed (and therefore M + N = M ∨ N). Orthogonality may be too strong an assumption, but it is sufficient to ensure the conclusion. It is known that something is necessary; if no additional assumptions are made, then M + N need not be closed (see [50, p. 28], and Problems 52–55 below). Here is the conclusion under another very strong but frequently usable additional assumption.
KeywordsHilbert Space Cardinal Number Linear Manifold Infinite Dimension Elegant Solution
Unable to display preview. Download preview PDF.