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matqkks

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matqkks

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HallsofIvy

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But inner products are also important in adding a

i) [itex]d(u, v)\ge 0[/itex] and d(u, v)= 0 if and only if u= v.

ii) [itex]d(u,v)\le d(u, w)+ d(w, v)[/itex] for any vectors, u, v, w.

We interpret d(u, v) as the distance between vectors u and v (a "metric vector space" in which Cauchy sequences converge is called a "Frechet" space). That doesn't really make use of the algebraic properties of a vector space but leads to a "normed space" which has a norm, a function that maps a single vector to a real number such that

i) [itex]|v|\ge 0[/itex] and |v|= 0 if and only if v= 0.

ii) [itex]|u+ v|\le |u|+ |v|[/itex].

iii) [itex]|\alpha v|= |\alpha| |v|[/itex] for any vector v and scalar [itex]\alpha[/itex] ([itex]|\alpha|[/itex] is the usual absolute value of [itex]\alpha[/itex]).

and we interpret |v| as the "length" of vector v. Of course, if we have a "length" we have a "distance" and can define d(u, v)= |u- v| so we have all the properties of a "metric vector space" plus additional ones. A normed space in which all Cauchy sequences of vectors converge is called a "Banach space". Other than the Euclidean spaces themselves, the most important example of a Banach space is L

We define an "inner product" on a vector space as a function that maps pairs of vectors to scalars (typically real or complex numbers), satisfying

i) [itex]<i, i>\ge 0[/itex] and <i, i>= 0 if and only if v= 0

ii) [itex]<\alpha u, v>= \alpha<u, v>[/itex] for any two vectors, u and v, and any scalar [itex]\alpha[/itex].

iii)[itex]<u+v, w>= <u, w>+ <v, w>[/itex]

Once we have an inner product we can define [itex]|v|= \sqrt{<v, v>}[/itex] and have all of the properties of a normed space (which includes all of the properties of a metric vector space) plus additional properties. An inner product space in whichy all Cauchy sequences of vectors converge is called a "Hilbert space". Other than the Euclidean space themselves, the most important Hilbert space is L

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matqkks

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Thanls for the detailed reply.

- #4

chiro

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With inner product spaces you can do projections which has not only a geometric importance, but an importance in terms of an abstract decomposition.

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