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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.37 Hilbert Spaces

The theory Of VECTOR SPACES [I.3 §2.3] and LINEAR MAPS [I.3 §4.2] underpins a large part of mathematics. However, angles cannot be defined using vector space concepts alone, since linear maps do not in general preserve angles. An inner product space can be thought of as a vector space with just enough extra structure for the notion of angle to make sense.

The simplest example of an inner product on a vector space is the standard scalar product defined on Imagen, the space of all real sequences of length n, as follows. If v = (v1, . . . , vn) and w = (W1,. . . , wn) are two such sequences, then their scalar product, denoted 〈 ...

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