By G. N. Watson

An Unabridged, Digitally Enlarged Printing, to incorporate: The Tabulation Of Bessel features - Bibliography - Index Of Symbols - record Of Authors Quoted, And A finished Index

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Thus S ⊥ ⊂ (span S)⊥ . 1. A collection of vectors S in an inner product space H is called orthonormal if 1 x=y∈S x, y = 0 x = y, x, y ∈ S. An orthonormal collection S is called an orthonormal basis if span S = H. 3. Let S be an orthonormal set of vectors in a Hilbert space H. Then the span S consists of all vectors of the form ∞ x= α j xj , xj ∈ S, j = 1, . . , ∞, j=1 10-1 ( ) 10-2 10. GEOMETRY OF A HILBERT SPACE AND GRAM-SCHMIDT PROCESS where the αj are square summable: ∞ |αj |2 < ∞. j=1 The sum converges in the Hilbert space: n x− αj xj → 0, j=1 and ∞ x 2 |αj |2 .

Continuity of pU follows since the sets [0, b), (b, ∞) as b ranges over (0, ∞) generate the topology on [0, ∞). Now, any topology under which every p ∈ S is continuous certainly contains the collection C = {p−1 [0, b) : b ∈ (0, ∞)}, consisting of convex, symmetric sets with the origin as an interior point. Consider the smallest TVS topology containing this collection. It is easily seen to be locally convex. ) Since p = pU for U = p−1 [0, 1) we see from the above argument that all p ∈ S are continuous in this topology.

Recall that a topology on X is a collection U of subsets of X that includes X and ∅, and is closed under unions and finite intersections. The intersection of a family of topologies is also a topology. Thus the weakest topology with property A is the intersection of all topologies with property A. Proof. Exercise. What are some examples of LCSs? First off, any Banach space is locally convex, since the open balls at the origin are a basis of convex sets. But not every LCS has a norm which is compatible with the topology.