Geometry of Hilbert Spaces

Abstract

Given a subset M of an inner product space V the orthogonal complement $M^{\perp}$ of M is the set of all vectors in V which are orthogonal to every vector in M. Elementary properties of the orthogonal complement are investigated first; then as a first basic result on orthogonal complements the projection theorem in Hilbert spaces is proven. The next section introduces Gram determinants and explains their geometrical meaning. With the help of the projection theorem it is straight forward to determine the topological dual of a Hilbert space. The Riesz-Fréchet theorem states that a Hilbert space and its dual are anti-isomorphic. Another useful consequence is the extension theorem. A simple example illustrates the use of Hilbert space methods in finding a tempered elementary solution of the differential operator I − Δ _n , Δ _n being the Laplace operator in ${\mathbb R}^n$.