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 ^{n}, the space of all real sequences of length *n*, as follows. If *v* = (*v*_{1}, . . . , *v*_{n}) and *w* = (*W*_{1},. . . , *w*_{n}) are two such sequences, then their scalar product, denoted 〈 ...

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