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By Seidenberg A.

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N-l}) defined as N-I '/:". 1· ," n ej21rn·I,;/N " -= IN y. k = 0,1, ... ,N - 1, n=() where x, x E 1'2 ({O, 1, ... 6. /5 FouruT Dllal Spaces - I}). 1 are gencralizpd ill this section to spaces other than £2(lR). ransforms in which the Planchen~l and Parseval relations hold. These pairs of spaces are called Fourier dual spaces and are isomorphic. he same. 2. 5) are quite general in that they hold for all pairs 'H and it of Fourier dual spaces. Plancherel For all f (0 'H. «71.. ) £"2({O,l, ... ,lV - I}) 21 £"2 ({O, 1, ...

6. 4The terms signal and function are used interchangeably. 5 As a convention, unspecified limits on integrations, summations, or sequences are implicitly defined to cover the entire domain associated with the function in question; for example, {

4 Operators 15 10. T is a topological isomorphism if T is bijective, T E B('HI,1i2), and T- 1 E B(1i2' 1i 1 ). Thus, both T and T-l are continuous linear mappings. liTJII1-(2 = IIJII'Hl' 11. T is an isometry if for all J E 'Hl, 12. T is a unitary map if it is linear, bijective, and an isometry. If T is unitary then T- I = T*. Compositions of topological isomorphisms are again topological isomorphisms. Two Hilbert spaces 'Hl and 1i2 that are related by a topological isomorphism are topologically equivalent.

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