## Applications of Functional Analysis and Operator Theory by V. Hutson By V. Hutson

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Additional info for Applications of Functional Analysis and Operator Theory

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22 Suppose that {tP,,} and {i/J J are orthonormal sets in a Hilbert space, and assume that {tP,,} is complete. Prove that {i/JJ is complete if 2 tP" - i/J,,11 < 1. 1 Introduction In the last chapter we tried to convince the reader that a complete normed vector space is a suitable setting for analysis, but we gave only a few examples of complete spaces of functions-vs'(O) with the sup norm and related spacesand no example of a Hilbert space offunctions. To confine the analysis always to «5'(Q), useful as this space is, will severely limit the range of problems which can be tackled, and in particular will rule out the application of the powerful results of the abstract theory of operators on a Hilbert space.

7 Theorem. With g as above, suppose that Un) is a sequence ofmeasurable functions from X to "JR.

1 Introduction In the last chapter we tried to convince the reader that a complete normed vector space is a suitable setting for analysis, but we gave only a few examples of complete spaces of functions-vs'(O) with the sup norm and related spacesand no example of a Hilbert space offunctions. To confine the analysis always to «5'(Q), useful as this space is, will severely limit the range of problems which can be tackled, and in particular will rule out the application of the powerful results of the abstract theory of operators on a Hilbert space.