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| An '''inner product space''' is a ] with additional structure, an '''inner product''' or '''scalar product''', which allows to talk about angles and lengths of vectors. Inner product spaces are generalizations of ]. | An '''inner product space''' is a ] with additional structure, an '''inner product''' or '''scalar product''', which allows to talk about angles and lengths of vectors. Inner product spaces are generalizations of ]. | ||
| Formally, an inner product space is a ] or ] vector space ''V'' together with a map ''f'' : ''V'' x ''V'' → ''F'' where ''F'' is the ground field (either '''R''' or '''C'''). We write <''x'', ''y''> instead of ''f''(''x'', ''y'') and require that the following axioms be satisfied: | Formally, an inner product space is a ] or ] vector space ''V'' together with a map ''f'' : ''V'' x ''V'' → ''F'' where ''F'' is the ground field (either '''R''' or '''C'''). We write <''x'', ''y''> instead of ''f''(''x'', ''y'') and require that the following axioms be satisfied: | ||
| #for any ''x'' in ''V'', <''x'', ''x''> ≥ 0 and <''x'', ''x''> = 0 if and only if ''x'' = 0 | #for any ''x'' in ''V'', <''x'', ''x''> ≥ 0 and <''x'', ''x''> = 0 if and only if ''x'' = 0 | ||
| #<''z'', ''ax+y''> = ''a'' <''z'', ''x''> + <''z'', ''y''> for any ''a'' in ''F'' and ''x'', ''y'' in ''V''. | #<''z'', ''ax+y''> = ''a'' <''z'', ''x''> + <''z'', ''y''> for any ''a'' in ''F'' and ''x'', ''y'' in ''V''. | ||
| #<''x'', ''y''> = <''y'', ''x''><sup>*</sup> whenever ''x'', ''y'' are in V. Here <sup>*</sup> denotes complex conjugation; if ''F'' = '''R''', we have <''x'', ''y''> = <''y'', ''x''>. | #<''x'', ''y''> = <''y'', ''x''><sup>*</sup> whenever ''x'', ''y'' are in V. Here <sup>*</sup> denotes complex conjugation; if ''F'' = '''R''', we have <''x'', ''y''> = <''y'', ''x''>. | ||
| A function which follows the second and third axioms is called a ''sesqui-linear operator'' | A function which follows the second and third axioms is called a ''sesqui-linear operator'' | ||
| (one-and-a-half linear operator). A sesqui-linear operator which is ''positive'' | (one-and-a-half linear operator). A sesqui-linear operator which is ''positive'' | ||
| (<''x'', ''x''> ≥ 0) is called a ''semi inner product''. | (<''x'', ''x''> ≥ 0) is called a ''semi inner product''. | ||
| Note that many authors require an inner product to be linear in the first and conjugate-linear in the second argument, contrary to the convention adopted above. This change is immaterial, but the definition above ensures a smoother connection to the ] popular in ]. | Note that many authors require an inner product to be linear in the first and conjugate-linear in the second argument, contrary to the convention adopted above. This change is immaterial, but the definition above ensures a smoother connection to the ] popular in ]. | ||
| Here and in the sequel, we will write ||''x''|| for √<''x'', ''x''>. This is well defined by axiom 1 and is thought of as the length of the vector ''x''. | Here and in the sequel, we will write ||''x''|| for √<''x'', ''x''>. This is well defined by axiom 1 and is thought of as the length of the vector ''x''. | ||
| From these axioms, we can conclude the following: | From these axioms, we can conclude the following: | ||
| ⚫ | *Theorem (]): |<''x'', ''y''>| ≤ ||''x''||·||''y''|| for any ''x'', ''y'' in ''V'' | ||
| ⚫ | *Theorem (Cauchy-Schwarz): |<''x'', ''y''>| ≤ ||''x''||·||''y''|| for any ''x'', ''y'' in ''V'' | ||
| *Theorem (]): ||''x'' + ''y''|| ≤ ||''x''|| + ||''y''|| | *Theorem (]): ||''x'' + ''y''|| ≤ ||''x''|| + ||''y''|| | ||
| *Theorem (Pythagoras): Whenever ''x'', ''y'' are in ''V'' and <''x'', ''y''> = 0, then ||x||<sup>2</sup> + ||y||<sup>2</sup> = ||x+y||<sup>2</sup>. | *Theorem (Pythagoras): Whenever ''x'', ''y'' are in ''V'' and <''x'', ''y''> = 0, then ||x||<sup>2</sup> + ||y||<sup>2</sup> = ||x+y||<sup>2</sup>. | ||
| An induction on Pythagoras yields: | An induction on Pythagoras yields: | ||
| *Theorem (Pythagoras): If ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> are orthogonal vectors, that is, <''x''<sub>''j''</sub>, ''x''<sub>''k''</sub>> = 0 whenever ''j'' ≠ ''k'', then | *Theorem (Pythagoras): If ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> are orthogonal vectors, that is, <''x''<sub>''j''</sub>, ''x''<sub>''k''</sub>> = 0 whenever ''j'' ≠ ''k'', then | ||
| :∑ ||''x''<sub>''k''</sub>||<sup>2</sup> = ||∑ ''x''<sub>''k''</sub>||<sup>2</sup> | :∑ ||''x''<sub>''k''</sub>||<sup>2</sup> = ||∑ ''x''<sub>''k''</sub>||<sup>2</sup> | ||
| Because of the triangle inequality and because of axiom 2, we see that ||·|| is a norm which turns ''V'' into a ] and hence also into a ]. The most important inner product spaces are the ones which are complete with respect to this metric; they are called ]. | Because of the triangle inequality and because of axiom 2, we see that ||·|| is a norm which turns ''V'' into a ] and hence also into a ]. The most important inner product spaces are the ones which are complete with respect to this metric; they are called ]. | ||
| In view of the Cauchy-Schwarz inequality, we also note that <·,·> is ] from ''V'' x ''V'' to ''F''. This allows us to extend Pythagoras' theorem a tiny bit more, and rename it: | In view of the Cauchy-Schwarz inequality, we also note that <·,·> is ] from ''V'' x ''V'' to ''F''. This allows us to extend Pythagoras' theorem a tiny bit more, and rename it: | ||
| *Theorem (Parseval's Identity): If ''x''<sub>''k''</sub> are mutually orthogonal vectors in ''V'' and if ∑ ''x''<sub>''k''</sub> converges to ''x'' in ''V'', then | *Theorem (Parseval's Identity): If ''x''<sub>''k''</sub> are mutually orthogonal vectors in ''V'' and if ∑ ''x''<sub>''k''</sub> converges to ''x'' in ''V'', then | ||
| :∑ ||''x''<sub>''k''</sub>||<sup>2</sup> = ||''x''||<sup>2</sup> | :∑ ||''x''<sub>''k''</sub>||<sup>2</sup> = ||''x''||<sup>2</sup> | ||
| Another consequence of the Cauchy-Schwarz inequality is that it is possible to define the ] φ between two non-zero vectors ''x'' and ''y'' (at least in the case ''F'' = '''R''') by writing | Another consequence of the Cauchy-Schwarz inequality is that it is possible to define the ] φ between two non-zero vectors ''x'' and ''y'' (at least in the case ''F'' = '''R''') by writing | ||
| :cos(φ) = <''x'', ''y''> / (||''x''||·||''y''||) | :cos(φ) = <''x'', ''y''> / (||''x''||·||''y''||) | ||
| in analogy to the situation in ]. | in analogy to the situation in ]. | ||
| Several types of maps ''A'' : ''V'' <tt>-></tt> ''W'' between inner product spaces are of relevance: | Several types of maps ''A'' : ''V'' <tt>-></tt> ''W'' between inner product spaces are of relevance: | ||
| * ], i.e. ''A''(''ax'' + ''y'') = ''a'' ''A''(''x'') + ''A''(''y'') for all ''a'' in ''F'' and all ''x'' and ''y'' in ''V''. | * ], i.e. ''A''(''ax'' + ''y'') = ''a'' ''A''(''x'') + ''A''(''y'') for all ''a'' in ''F'' and all ''x'' and ''y'' in ''V''. | ||
| ⚫ | * ] linear maps, i.e. ''A'' is linear and continuous with respect to the metric defined above, or equivalently, ''A'' is linear and the set { ||''Ax''|| : ''x'' in ''V'' with ||''x''|| ≤ 1 } is bounded. | ||
| ⚫ | * Continuous linear maps, i.e. ''A'' is linear and continuous with respect to the metric defined above, or equivalently, ''A'' is linear and the set { ||''Ax''|| : ''x'' in ''V'' with ||''x''|| ≤ 1 } is bounded. | ||
| * Isometries, i.e. ''A'' is linear and <''Ax'', ''Ay''> = <''x'', ''y''> for all ''x'', ''y'' in ''V'', or equivalently, ''A'' is linear and ||''Ax''|| = ||''x''|| for all ''x'' in ''V''. | * Isometries, i.e. ''A'' is linear and <''Ax'', ''Ay''> = <''x'', ''y''> for all ''x'', ''y'' in ''V'', or equivalently, ''A'' is linear and ||''Ax''|| = ||''x''|| for all ''x'' in ''V''. | ||
| * Isometrical isomorphism, i.e. ''A'' is a surjective isometry. | * Isometrical isomorphism, i.e. ''A'' is a surjective isometry. | ||
| From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. | From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. | ||
Revision as of 06:52, 7 February 2002
An inner product space is a vector space with additional structure, an inner product or scalar product, which allows to talk about angles and lengths of vectors. Inner product spaces are generalizations of Euclidean space.
Formally, an inner product space is a real or complex vector space V together with a map f : V x V → F where F is the ground field (either R or C). We write <x, y> instead of f(x, y) and require that the following axioms be satisfied:
- for any x in V, <x, x> ≥ 0 and <x, x> = 0 if and only if x = 0
- <z, ax+y> = a <z, x> + <z, y> for any a in F and x, y in V.
- <x, y> = <y, x> whenever x, y are in V. Here denotes complex conjugation; if F = R, we have <x, y> = <y, x>.
A function which follows the second and third axioms is called a sesqui-linear operator (one-and-a-half linear operator). A sesqui-linear operator which is positive (<x, x> ≥ 0) is called a semi inner product.
Note that many authors require an inner product to be linear in the first and conjugate-linear in the second argument, contrary to the convention adopted above. This change is immaterial, but the definition above ensures a smoother connection to the bra-ket notation popular in quantum mechanics.
Here and in the sequel, we will write ||x|| for √<x, x>. This is well defined by axiom 1 and is thought of as the length of the vector x.
From these axioms, we can conclude the following:
- Theorem (Cauchy-Schwarz): |<x, y>| ≤ ||x||·||y|| for any x, y in V
- Theorem (Triangle Inequality): ||x + y|| ≤ ||x|| + ||y||
- Theorem (Pythagoras): Whenever x, y are in V and <x, y> = 0, then ||x|| + ||y|| = ||x+y||.
An induction on Pythagoras yields:
- Theorem (Pythagoras): If x1, ..., xn are orthogonal vectors, that is, <xj, xk> = 0 whenever j ≠ k, then
- ∑ ||xk|| = ||∑ xk||
Because of the triangle inequality and because of axiom 2, we see that ||·|| is a norm which turns V into a normed vector space and hence also into a metric space. The most important inner product spaces are the ones which are complete with respect to this metric; they are called Hilbert spaces.
In view of the Cauchy-Schwarz inequality, we also note that <·,·> is continuous from V x V to F. This allows us to extend Pythagoras' theorem a tiny bit more, and rename it:
- Theorem (Parseval's Identity): If xk are mutually orthogonal vectors in V and if ∑ xk converges to x in V, then
- ∑ ||xk|| = ||x||
Another consequence of the Cauchy-Schwarz inequality is that it is possible to define the angle φ between two non-zero vectors x and y (at least in the case F = R) by writing
- cos(φ) = <x, y> / (||x||·||y||)
in analogy to the situation in Euclidean space.
Several types of maps A : V -> W between inner product spaces are of relevance:
- Linear maps, i.e. A(ax + y) = a A(x) + A(y) for all a in F and all x and y in V.
- Continuous linear maps, i.e. A is linear and continuous with respect to the metric defined above, or equivalently, A is linear and the set { ||Ax|| : x in V with ||x|| ≤ 1 } is bounded.
- Isometries, i.e. A is linear and <Ax, Ay> = <x, y> for all x, y in V, or equivalently, A is linear and ||Ax|| = ||x|| for all x in V.
- Isometrical isomorphism, i.e. A is a surjective isometry.
From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic.