|
|
| Line 1: |
Line 1: |
| {{For|the scalar product or dot product of coordinate vectors|dot product}}
| | Although some of the methods he tried gave him relief at first, the ringing noise usually comes back after a few days; and the only thing he can think of to treat his tinnitus was to undergo a surgery. The upside to these remedies is that they are safe with no health risk involved. By eliminating other sources of stress, you better equip yourself to deal with tinnitus problems. Treating the blood pressure usually will relieve the tinnitus. Counseling can be beneficial in children who have developed phonophobia. <br><br>Misophonia is a rare and little understood sound sensitivity disorder that is characterized by an extreme aversion to selective sounds. In these people, relaxation and therefore the reduction of stress have proven to be very effective means of treatment. It calls for blocking out the noise with a "masker" that you put on your ear to block out the disturbing noise and, at the same time, trying to concentrate on other noises than those that are in their ears. For a variety of reasons, you may at some point start hearing a ringing or other type of sound in your ears. This cure will make your problem vanish slowly, if not soon. <br><br>The problem with tinnitus is that there arent any effective medicines or drugs that can cure your problem. Our bodily functions produce sounds constantly, but we do not generally hear these as they are masked by outside noises. Another problem is that even everyday drug treatments could cause some individuals a problem with tinnitus. Based on this information, lipoflavonoid for tinnitus can be effective. But those entire action just results to nothing at all. <br><br>I have sat right next to a massive speaker during a concert and have found out my ears were ringing almost immediately after the concert and sometimes even during the concert. Ear, Nose, and Throat (ENT) specialists can positively diagnose the origin of the tinnitus that you are experiencing. s not possible to give the best solution to the problem and find the best way to clear tinnitus if you don. In wieweit chronische Intoxikationen beispielsweise mit Schwermetallen zur Ohrgeräuschentstehung beitragen muss derzeit noch als unklar bezeichnet werden. By looking into making some simple lifestyle modifications, adding one or two supplements, evaluating current medications, and practicing stress reduction techniques, sufferers can find that they can reduce or eliminate their symptoms with natural remedies, even the absence of a be-all and end-all medical cure. <br><br>The holistic therapy is based on scientific fact along with the study he completed. Doing exercises is another commonly used technique for curing tinnitus. Those suffering from nagging tinnitus can benefit from internet-based therapy just as much as patients who take part in group therapy sessions, notes the March 7, 2012 news release, "Internet-based therapy relieves persistent tinnitus. Get the best quality, organic essential oils you can find since they aren't very expensive and go a very long way. In studies carried out in the USA there was a big difference in the experiences of tinnitus sufferers who had good diets compared to those who ate a lot of junk food.<br><br>If you have any concerns relating to where and the best ways to use tinnitus diet, [http://www.dancefire.info/ www.dancefire.info],, you could contact us at our own web page. |
| {{See also|Parallelogram law|Polarization identity}}
| |
| [[Image:Inner-product-angle.png|thumb|300px|Geometric interpretation of the angle between two vectors defined using an inner product]]
| |
| In [[linear algebra]], an '''inner product space''' is a [[vector space]] with an additional [[Mathematical structure|structure]] called an '''inner product'''. This additional structure associates each pair of vectors in the space with a [[Scalar (mathematics)|scalar]] quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the [[length]] of a vector or the [[angle]] between two vectors. They also provide the means of defining [[orthogonality]] between vectors (zero inner product). Inner product spaces generalize [[Euclidean space]]s (in which the inner product is the [[dot product]], also known as the scalar product) to vector spaces of any (possibly infinite) [[dimension (vector space)|dimension]], and are studied in [[functional analysis]].
| |
| | |
| An inner product naturally induces an associated [[Norm (mathematics)|norm]], thus an inner product space is also a [[normed vector space]]. A [[complete space]] with an inner product is called a [[Hilbert space]]. An incomplete space with an inner product is called a '''pre-Hilbert space''', since its [[Complete space#Completion|completion]] with respect to the norm induced by the inner product becomes a [[Hilbert space]]. Inner product spaces over the field of complex numbers are sometimes referred to as '''unitary spaces.'''
| |
| | |
| == Definition ==
| |
| In this article, the [[field (mathematics)|field]] of [[scalar (mathematics)|scalar]]s denoted <math>F</math> is either
| |
| the field of [[real number]]s <math>\R</math> or the field of [[complex number]]s <math>\mathbb{C}</math>.
| |
| | |
| Formally, an inner product space is a [[vector space]] ''V'' [[Algebra over a field|over the field]] <math>F</math> together with an ''inner product'', i.e., with a map
| |
| :<math> \langle \cdot, \cdot \rangle : V \times V \rightarrow F </math>
| |
| | |
| that satisfies the following three [[axiom]]s for all vectors <math>x,y,z \in V</math> and all scalars <math>a \in F</math>:<ref name= Jain>
| |
| | |
| {{cite book |title=Functional analysis |author=P. K. Jain, Khalil Ahmad |url=http://books.google.com/?id=yZ68h97pnAkC&pg=PA203 |page=203 |chapter=5.1 Definitions and basic properties of inner product spaces and Hilbert spaces |isbn=81-224-0801-X |year=1995 |edition=2nd |publisher=New Age International}}
| |
| | |
| </ref><ref name="Prugovec̆ki">
| |
| | |
| {{cite book |title=Quantum mechanics in Hilbert space |author=Eduard Prugovec̆ki |url=http://books.google.com/?id=GxmQxn2PF3IC&pg=PA18 |chapter=Definition 2.1 |pages=18 ''ff'' |isbn=0-12-566060-X |year=1981 |publisher=Academic Press |edition =2nd}}
| |
| | |
| </ref>
| |
| | |
| * [[complex conjugate|Conjugate]] symmetry:
| |
| | |
| ::<math>\langle x,y\rangle =\overline{\langle y,x\rangle}.</math>
| |
| Note that when <math>F = \R</math>, there is symmetry.
| |
| | |
| * [[Linear]]ity in the first argument:
| |
| | |
| ::<math>\langle ax,y\rangle= a \langle x,y\rangle.</math>
| |
| ::<math>\langle x+y,z\rangle= \langle x,z\rangle+ \langle y,z\rangle.</math>
| |
| | |
| * [[Definite bilinear form|Positive-definiteness]]:
| |
| | |
| ::<math>\langle x,x\rangle \geq 0</math>
| |
| :: <math>\langle x,x\rangle = 0 \Rightarrow x = 0</math>
| |
| | |
| === Alternative definition and remarks===
| |
| | |
| Some authors, especially in [[physics]] and [[matrix algebra]], prefer to define the inner product and the [[sesquilinear form]] with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second.
| |
| In those disciplines we would write the product <math>\langle x,y\rangle</math> as <math>\langle y|x\rangle</math> (the [[bra-ket notation]] of [[quantum mechanics]]), respectively <math>y^\dagger x</math> (dot product as a case of the convention of forming the matrix product ''AB'' as the dot products of rows of ''A'' with columns of ''B''). Here the kets and columns are identified with the vectors of ''V'' and the bras and rows with the [[dual vector]]s or [[linear functional]]s of the [[dual space]] ''V''<sup>∗</sup>, with conjugacy associated with duality. This reverse order is now occasionally followed in the more abstract literature,<ref>{{cite book|last=Emch|first=Gerard G.|title=Algebraic methods in statistical mechanics and quantum field theory|year=1972|publisher=[[Wiley-Interscience]]|location=New York|isbn=978-0-471-23900-0}}</ref> taking <math>\langle x,y\rangle</math> to be conjugate linear in ''x'' rather than ''y''. A few instead find a middle ground by recognizing both <math>\langle , \rangle</math> and <math>\langle | \rangle</math> as distinct notations differing only in which argument is conjugate linear.
| |
| | |
| There are various technical reasons why it is necessary to restrict the [[basefield]] to <math>\R</math> and <math>\C</math> in the definition. Briefly, the basefield has to contain an [[ordered field|ordered subfield]]{{Citation needed|date=June 2010}} (in order for non-negativity to make sense) and therefore has to have [[characteristic (algebra)|characteristic]] equal to 0 (since any ordered field has to have such characteristic). This immediately excludes finite fields. The basefield has to have additional structure, such as a distinguished automorphism. More generally any quadratically closed subfield of <math>\R</math> or <math>\mathbb{C}</math> will suffice for this purpose, e.g., the [[algebraic number]]s, but when it is a proper subfield (i.e., neither <math>\R</math> nor <math>\mathbb{C}</math>) even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over <math>\R</math> or <math>\mathbb{C}</math>, such as those used in [[quantum computation]], are automatically [[Complete metric space|metrically complete]] and hence [[Hilbert space]]s.
| |
| | |
| In some cases we need to consider non-negative ''semi-definite'' sesquilinear forms. This means that <math>\langle x,x\rangle</math> is only required to be non-negative. We show how to treat these below.
| |
| | |
| === Elementary properties ===
| |
| Notice that conjugate symmetry implies that <math>\langle x,x \rangle</math> is real for all <math>x</math>, since we have <math>\langle x,x \rangle = \overline{\langle x,x \rangle}.</math>
| |
| | |
| Moreover, sesquilinearity (see below) implies that <math> \langle -x,x \rangle= -1\langle x,x\rangle = \overline{-1}\langle x,x\rangle = \langle x,-x\rangle.</math>
| |
| | |
| Conjugate symmetry and linearity in the first variable gives
| |
| :<math> \langle x, a y \rangle = \overline{\langle a y, x \rangle} = \overline{a} \overline{\langle y, x \rangle} = \overline{a} \langle x, y \rangle </math>
| |
| | |
| :<math> \langle x, y + z \rangle = \overline{\langle y + z, x \rangle} = \overline{\langle y, x \rangle} + \overline{\langle z, x \rangle} = \langle x, y \rangle + \langle x, z \rangle,</math>
| |
| | |
| so an inner product is a [[sesquilinear]] form.
| |
| Conjugate symmetry is also called Hermitian symmetry, and a conjugate symmetric sesquilinear form is called a ''Hermitian form''.
| |
| While the above axioms are more mathematically economical, a compact verbal definition of an inner product is a ''positive-definite Hermitian form''.
| |
| | |
| In the case of <math> F = \R </math>, conjugate-symmetry reduces to symmetry, and sesquilinear reduces to bilinear.
| |
| So, an inner product on a real vector space is a ''positive-definite symmetric bilinear form''.
| |
| | |
| From the linearity property it is derived that <math>x = 0</math> implies <math>\langle x,x \rangle = 0,</math> while from the positive-definiteness axiom we obtain the converse, <math>\langle x,x \rangle = 0</math> implies <math>x = 0.</math>
| |
| Combining these two, we have the property that <math>\langle x,x \rangle = 0</math> if and only if <math>x = 0.</math>
| |
| | |
| Combining the linearity of the inner product in its first argument and the conjugate symmetry gives the following important generalization of the familiar square expansion:
| |
| ::<math>\langle x + y,x + y\rangle = \langle x,x\rangle + \langle x,y\rangle + \langle y,x\rangle + \langle y,y\rangle.</math>
| |
| Assuming that '''the underlying field is''' <math>\R</math>, the inner product becomes symmetric, and we obtain
| |
| ::<math>\langle x + y,x + y\rangle =\langle x,x\rangle + 2\langle x,y\rangle + \langle y,y\rangle,</math>
| |
| or similarly,
| |
| ::<math>\langle x - y,x - y\rangle =\langle x,x\rangle - 2\langle x,y\rangle + \langle y,y\rangle.</math>
| |
| | |
| The property of an inner product space <math> V </math> that
| |
| ::<math> \langle x+y,z\rangle= \langle x,z\rangle+ \langle y,z\rangle </math> and <math> \langle x,y+z\rangle = \langle x,y\rangle + \langle x,z\rangle </math>
| |
| is also known as ''additivity''.
| |
| | |
| == Examples ==
| |
| * A simple example is the [[real numbers]] with the standard multiplication as the inner product
| |
| ::<math>\langle x,y\rangle := x y.</math>
| |
| :More generally, the [[real coordinate space|real ''n''-space]] <math>\mathbb{R}</math><sup>''n''</sup> with the [[dot product]] is an inner product space, an example of a [[Euclidean space|Euclidean ''n''-space]].
| |
| ::<math>\langle (x_1,\ldots, x_n),(y_1,\ldots, y_n)\rangle := x^\mathsf{T} y = \sum_{i=1}^{n} x_i y_i = x_1 y_1 + \cdots + x_n y_n,</math>
| |
| :where ''x''<sup>T</sup> is the [[transpose]] of ''x''.
| |
| | |
| *The general form of an inner product on <math>\mathbb{C}</math><sup>''n''</sup> is known as the [[Hermitian form]] and is given by
| |
| ::<math>\langle \mathbf{x},\mathbf{y}\rangle := \mathbf{y}^\dagger\mathbf{M}\mathbf{x} = \overline{\mathbf{x}^\dagger\mathbf{M}\mathbf{y}},</math>
| |
| :where '''M''' is any [[Hermitian matrix|Hermitian]] [[positive-definite matrix]] and '''y'''<sup>†</sup> is the [[conjugate transpose]] of '''y'''. For the real case this corresponds to the dot product of the results of directionally different [[Scaling (geometry)|scaling]] of the two vectors, with positive [[scale factor]]s and orthogonal directions of scaling. Up to an orthogonal transformation it is a [[Weight function|weighted-sum]] version of the dot product, with positive weights.
| |
| | |
| *The article on [[Hilbert space]] has several examples of inner product spaces wherein the metric induced by the inner product yields a [[complete space|complete]] metric space. An example of an inner product which induces an incomplete metric occurs with the space ''C''([''a'', ''b'']) of continuous complex valued functions on the interval [''a'', ''b'']. The inner product is
| |
| ::<math> \langle f , g \rangle := \int_a^b f(t) \overline{g(t)} \, dt. </math>
| |
| :This space is not complete; consider for example, for the interval {{closed-closed|−1,1}} the sequence of continuous "step" functions { ''f''<sub>''k''</sub> }<sub>''k''</sub> where
| |
| :* ''f''<sub>''k''</sub>(''t'') is 0 for ''t'' in the subinterval {{closed-closed|−1,0}}
| |
| :* ''f''<sub>''k''</sub>(''t'') is 1 for ''t'' in the subinterval {{closed-closed|1/''k'', 1}}
| |
| :* ''f''<sub>''k''</sub> is [[affine transformation|affine]] in {{open-open|0, 1/''k''}}. That is, ''f''<sub>''k''</sub>(''t'') = ''kt''.
| |
| :This sequence is a [[Cauchy sequence]] for the norm induced by the preceding inner product, which does not converge to a ''continuous'' function.
| |
| | |
| *For real [[random variable]]s ''X'' and ''Y'', the [[expected value]] of their product
| |
| ::<math> \langle X, Y \rangle := \operatorname{E}(X Y) </math>
| |
| :is an inner product. In this case, {{nowrap|<''X'', ''X''> {{=}} 0}} if and only if [[probability|Pr]](''X'' = 0) = 1 (i.e., {{nowrap|''X'' {{=}} 0}} [[almost surely]]). This definition of expectation as inner product can be extended to [[random vector]]s as well.
| |
| | |
| *For square real matrices, <math>\langle A, B \rangle := \mathrm{tr}(AB^\mathsf{T})</math> with transpose as conjugation <math> \big(\langle A, B \rangle = \langle B^\mathsf{T}, A^\mathsf{T} \rangle \big)</math> is an inner product.
| |
| | |
| == Norms on inner product spaces ==<!-- This section is linked from [[Cauchy–Schwarz inequality]] -->
| |
| A linear space with a norm such as:
| |
| :<math>\|x\|_p = \left[ \sum_{i=1}^{\infty} |\xi_i|^p \right] ^{1/p} \ x = \{\xi_i\} \in \mathit {l}^p \ ,</math>
| |
| where ''p'' ≠ 2 is a [[normed space]] but not an inner product space, because this norm does not satisfy the [[Parallelogram_equality#Normed_vector_spaces_satisfying_the_parallelogram_law|parallelogram equality]] required of a norm to have an inner product associated with it.<ref name=Ahmed>
| |
| | |
| {{cite book |title=''Cited work'' |author=P. K. Jain, Khalil Ahmad |url=http://books.google.com/?id=yZ68h97pnAkC&pg=PA209 |page=209 |chapter=Example 5 |isbn=81-224-0801-X |year=1995 }}
| |
| | |
| </ref><ref name=Saxe>
| |
| | |
| {{cite book |title=Beginning functional analysis |author=Karen Saxe |url=http://books.google.com/?id=QALoZC64ea0C&pg=PA7 |page=7 |isbn=0-387-95224-1 |year=2002 |publisher=Springer}}
| |
| </ref> | |
| | |
| However, inner product spaces have a naturally defined [[Norm (mathematics)|norm]] based upon the inner product of the space itself that does satisfy the parallelogram equality:
| |
| | |
| :<math> \|x\| =\sqrt{\langle x, x\rangle}.</math>
| |
| | |
| This is well defined by the nonnegativity axiom of the definition of inner product space. The norm is thought of as the length of the vector ''x''.
| |
| Directly from the axioms, we can prove the following:
| |
| | |
| *[[Cauchy–Schwarz inequality]]: for ''x'', ''y'' elements of ''V''
| |
| | |
| :: <math> |\langle x,y\rangle| \leq \|x\| \cdot \|y\| </math>
| |
| | |
| :with equality if and only if ''x'' and ''y'' are [[linearly independent|linearly dependent]]. This is one of the most important inequalities in mathematics. It is also known in the Russian mathematical literature as the ''Cauchy–Bunyakovsky–Schwarz inequality''.
| |
| | |
| :Because of its importance, its short proof should be noted.
| |
| | |
| ::It is trivial to prove the inequality true in the case ''y'' = 0. Thus we assume <math>\langle y, y \rangle</math> is nonzero, giving us the following:
| |
| | |
| ::<math> \lambda = \langle y , y \rangle^{-1} \langle x, y\rangle</math>
| |
| ::<math> 0 \leq \langle x -\lambda y, x -\lambda y \rangle = \langle x, x\rangle - \langle y , y \rangle^{-1} | \langle x,y\rangle|^2. </math>
| |
| | |
| ::The complete proof can be obtained by multiplying out this result.
| |
| | |
| *[[Orthogonal]]ity: The geometric interpretation of the inner product in terms of angle and length, motivates much of the geometric terminology we use in regard to these spaces. Indeed, an immediate consequence of the Cauchy–Schwarz inequality is that it justifies defining the [[angle]] between two non-zero vectors ''x'' and ''y'' in the case <math>F = \mathbb{R}</math> by the identity
| |
| | |
| :<math>\operatorname{angle}(x,y) = \arccos \frac{\langle x, y \rangle}{\|x\| \cdot \|y\|}.</math>
| |
| | |
| :We assume the value of the angle is chosen to be in the interval {{closed-closed|0, +π}}. This is in analogy to the situation in two-dimensional [[Euclidean space]].
| |
| | |
| :In the case <math>F = \mathbb{C}</math>, the angle in the interval {{closed-closed|0, +π/2}} is typically defined by
| |
| | |
| :<math>\operatorname{angle}(x,y) = \arccos \frac{|\langle x, y \rangle|}{\|x\| \cdot \|y\|}.</math>
| |
| | |
| :Correspondingly, we will say that non-zero vectors ''x'' and ''y'' of ''V'' are orthogonal if and only if their inner product is zero.
| |
| | |
| *[[homogeneous function|Homogeneity]]: for ''x'' an element of ''V'' and ''r'' a scalar
| |
| | |
| ::<math> \|r \cdot x\| = |r| \cdot \| x\|.</math>
| |
| | |
| :The homogeneity property is completely trivial to prove.
| |
| | |
| *[[Triangle inequality]]: for ''x'', ''y'' elements of ''V''
| |
| | |
| ::<math> \|x + y\| \leq \|x \| + \|y\|. </math>
| |
| | |
| :The last two properties show the function defined is indeed a norm.
| |
| | |
| :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 [[completeness (topology)|complete]] with respect to this metric; they are called [[Hilbert space]]s. Every inner product ''V'' space is a [[Dense set|dense]] subspace of some Hilbert space. This Hilbert space is essentially uniquely determined by ''V'' and is constructed by completing ''V''.
| |
| | |
| *[[Pythagorean theorem]]: Whenever ''x'', ''y'' are in ''V'' and ⟨''x'', ''y''⟩ = 0, then
| |
| | |
| ::<math> \|x\|^2 + \|y\|^2 = \|x+y\|^2. </math>
| |
| | |
| :The proof of the identity requires only expressing the definition of norm in terms of the inner product and multiplying out, using the property of additivity of each component.
| |
| | |
| :The name ''Pythagorean theorem'' arises from the geometric interpretation of this result as an analogue of the theorem in [[synthetic geometry]]. Note that the proof of the Pythagorean theorem in synthetic geometry is considerably more elaborate because of the paucity of underlying structure. In this sense, the synthetic Pythagorean theorem, if correctly demonstrated, is deeper than the version given above.
| |
| | |
| :An [[mathematical induction|induction]] on the Pythagorean theorem yields:
| |
| | |
| *If ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> are [[orthogonal]] vectors, that is, <math>\langle x_j,x_k\rangle=0</math> for distinct indices ''j'', ''k'', then
| |
| | |
| ::<math> \sum_{i=1}^n \|x_i\|^2 = \left\|\sum_{i=1}^n x_i \right\|^2. </math>
| |
| | |
| :In view of the Cauchy-Schwarz inequality, we also note that <math>\langle\cdot,\cdot\rangle</math> is [[continuous function|continuous]] from {{nowrap|''V'' × ''V''}} to ''F''. This allows us to extend Pythagoras' theorem to infinitely many summands:
| |
| | |
| *Parseval's identity: Suppose ''V'' is a ''complete'' inner product space. If {''x''<sub>''k''</sub>} are mutually orthogonal vectors in ''V'' then
| |
| | |
| ::<math> \sum_{i=1}^\infty\|x_i\|^2 = \left\|\sum_{i=1}^\infty x_i\right\|^2, </math>
| |
| | |
| :''provided the infinite series on the left is [[Convergence (series)|convergent]].'' Completeness of the space is needed to ensure that the sequence of partial sums
| |
| | |
| ::<math> S_k = \sum_{i=1}^k x_i, </math>
| |
| | |
| :which is easily shown to be a [[Cauchy sequence]], is convergent.
| |
| | |
| *[[Parallelogram law]]: for ''x'', ''y'' elements of ''V'',
| |
| | |
| ::<math> \|x + y\|^2 + \|x - y\|^2 = 2\|x\|^2 + 2\|y\|^2. </math>
| |
| | |
| The Parallelogram law is, in fact, a necessary and sufficient condition for the existence of a scalar
| |
| product corresponding to a given norm. If it holds, the scalar product is defined by the
| |
| [[polarization identity]]:
| |
| | |
| ::<math> \|x + y\|^2 = \|x\|^2 + \|y\|^2 + 2 \real \langle x , y \rangle. </math>
| |
| | |
| :which is a form of the [[law of cosines]].
| |
| | |
| == Orthonormal sequences ==
| |
| Let ''V'' be a finite dimensional inner product space of dimension ''n''. Recall that every [[Basis (linear algebra)|basis]] of ''V'' consists of exactly ''n'' linearly independent vectors. Using the [[Gram–Schmidt process]] we may start with an arbitrary basis and transform it into an orthonormal basis. That is, into a basis in which all the elements are orthogonal and have unit norm. In symbols, a basis <math>\textstyle {\{e_1,\ldots,e_n\}}</math> is orthonormal if <math>\textstyle \langle e_i, e_j\rangle=0</math> if <math>\textstyle i\neq j</math> and <math>\textstyle \langle e_i, e_i\rangle = ||e_i|| = 1</math> for each ''i''.
| |
| | |
| This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way. Let ''V'' be any inner product space. Then a collection <math>\textstyle E=\{e_{\alpha}\}_{\alpha \in A}</math> is a ''basis'' for ''V'' if the subspace of ''V'' generated by finite linear combinations of elements of ''E'' is dense in ''V'' (in the norm induced by the inner product). We say that ''E'' is an ''orthonormal basis'' for ''V'' if it is a basis and <math>\textstyle \langle e_{\alpha}, e_{\beta}\rangle=0</math> if <math>\textstyle \alpha \neq \beta</math> and <math>\textstyle \langle e_{\alpha}, e_{\alpha}\rangle = ||e_{\alpha}|| = 1</math> for all <math>\textstyle \alpha,\beta \in A</math>.
| |
| | |
| Using an infinite-dimensional analog of the Gram-Schmidt process one may show:
| |
| | |
| '''Theorem'''. Any [[separable space|separable]] inner product space ''V'' has an orthonormal basis.
| |
| | |
| Using the [[Hausdorff maximal principle]] and the fact that in a [[Hilbert space|complete inner product space]] orthogonal projection onto linear subspaces is well-defined, one may also show that
| |
| | |
| '''Theorem'''. Any [[Hilbert space|complete inner product space]] ''V'' has an orthonormal basis.
| |
| | |
| The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis. The answer, it turns out is negative. This is a non-trivial result, and is proved below. The following proof is taken from Halmos's A Hilbert Space Problem Book (see the references). | |
| | |
| :{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
| |
| !Proof
| |
| |-
| |
| | Recall that the dimension of an inner product space is the cardinality of a maximal orthonormal system that it contains (by Zorn's lemma it contains at least one, and any two have the same cardinality). An orthonormal basis is certainly a maximal orthonormal system, but as we shall see, the converse need not hold. Observe that if ''G'' is a dense subspace of an inner product space ''H'', then any orthonormal basis for ''G'' is automatically an orthonormal basis for ''H''. Thus, it suffices to construct an inner product space space ''H'' with a dense subspace ''G'' whose dimension is strictly smaller than that of ''H''.
| |
| | |
| Let ''K'' be a Hilbert space of dimension <math>\aleph_0</math> (for instance, <math>K=\ell^2(\mathbb{N})</math>). Let ''E'' be an orthonormal basis of ''K'', so <math>|E| = \aleph_0</math>. Extend ''E'' to a [[Basis (linear algebra)#Related notions|Hamel basis]] <math>E \cup F</math> for ''K'', where <math>E \cap F = \emptyset</math>. Since it is known that the Hamel dimension of ''K'' is ''c'', the cardinality of the continuum, it must be that <math>|F| = c</math>.
| |
| | |
| Let ''L'' be a Hilbert space of dimension ''c'' (for instance, <math>L = \ell^2(\mathbb{R})</math>). Let ''B'' be an orthonormal basis for ''L'', and let <math>\phi : F \to B</math> be a bijection. Then there is a linear transformation <math>T : K \to L</math> such that <math>Tf = \phi(f)</math> for <math>f \in F</math>, and <math>Te = 0</math> for <math>e \in E</math>.
| |
| | |
| Let <math>H = K \oplus L</math> and let <math>G = \{(k,Tk) : k \in K)\}</math> be the graph of ''T''. Let <math>\bar{G}</math> be the closure of ''G'' in ''H''; we will show <math>\bar{G} = H</math>. Since for any <math>e \in E</math> we have <math>(e,0) \in G</math>, it follows that <math>K \oplus 0 \subset \bar{G}</math>.
| |
| | |
| Next, if <math>b \in B</math>, then <math>b = Tf</math> for some <math>f \in F \subset K</math>, so <math>(f,b) \in G \subset \bar{G}</math>; since <math>(f,0) \in \bar{G}</math> as well, we also have <math>(0,b) \in \bar{G}</math>. It follows that <math>0 \oplus L \subset \bar{G}</math>, so <math>\bar{G} = H</math>, and ''G'' is dense in ''H''.
| |
| | |
| Finally, <math>\{(e,0) : e \in E\}</math> is a maximal orthonormal set in ''G''; if
| |
| :<math>0 = \langle (e,0), (k, Tk) \rangle = \langle e,k \rangle + \langle 0,Tk \rangle = \langle e,k \rangle</math>
| |
| for all <math>e \in E</math> then certainly <math>k=0</math>, so <math>(k,Tk)=(0,0)</math> is the zero vector in ''G''. Hence the dimension of ''G'' is <math>|E| = \aleph_0</math>, whereas it is clear that the dimension of ''H'' is ''c''. This completes the proof.
| |
| |}
| |
| | |
| [[Parseval's identity]] leads immediately to the following theorem:
| |
| | |
| '''Theorem'''. Let ''V'' be a separable inner product space and {''e''<sub>''k''</sub>}<sub>''k''</sub> an orthonormal basis of ''V''.
| |
| Then the map
| |
| :<math> x \mapsto \{\langle e_k, x\rangle\}_{k \in \mathbb{N}} </math>
| |
| is an isometric linear map ''V'' → ''ℓ''<sup> 2</sup> with a dense image.
| |
| | |
| This theorem can be regarded as an abstract form of [[Fourier series]], in which an arbitrary orthonormal basis plays the role of the sequence of [[trigonometric polynomial]]s. Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided ''ℓ''<sup> 2</sup> is defined appropriately, as is explained in the article [[Hilbert space]]).
| |
| In particular, we obtain the following result in the theory of Fourier series:
| |
| | |
| '''Theorem'''. Let ''V'' be the inner product space <math>C[-\pi,\pi]</math>. Then the sequence (indexed on set of all integers) of continuous functions
| |
| | |
| :<math>e_k(t) = {1 \over \sqrt{2 \pi}}e^{i k t}</math>
| |
| | |
| is an orthonormal basis of the space <math>C[-\pi,\pi]</math> with the ''L''<sup>2</sup> inner product. The mapping
| |
| | |
| :<math> f \mapsto \frac{1}{\sqrt{2 \pi}} \left\{\int_{-\pi}^\pi f(t) e^{-i k t} \, dt \right\}_{k \in \mathbb{Z}} </math>
| |
| | |
| is an isometric linear map with dense image.
| |
| | |
| Orthogonality of the sequence {''e''<sub>''k''</sub>}<sub>''k''</sub> follows immediately from the fact that if ''k'' ≠ ''j'', then
| |
| | |
| :<math> \int_{-\pi}^\pi e^{-i (j-k) t} \, dt = 0. </math>
| |
| | |
| Normality of the sequence is by design, that is, the coefficients are so chosen so that the norm comes out to 1. Finally the fact that the sequence has a dense algebraic span, in the ''inner product norm'', follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous periodic functions on <math>[-\pi,\pi]</math> with the uniform norm. This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials.
| |
| | |
| ==Operators on inner product spaces==
| |
| Several types of [[linear]] maps ''A'' from an inner product space ''V'' to an inner product space ''W'' are of relevance:
| |
| * [[Continuous linear operator|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 of non-negative reals {||''Ax''||}, where ''x'' ranges over the closed unit ball of ''V'', is bounded.
| |
| * Symmetric linear operators, i.e., ''A'' is linear and <math>\langle Ax, y\rangle =\langle x, Ay\rangle</math> for all ''x'', ''y'' in ''V''.
| |
| * Isometries, i.e., ''A'' is linear and <math>\langle Ax, Ay\rangle =\langle x, y\rangle</math> for all ''x'', ''y'' in ''V'', or equivalently, ''A'' is linear and ||''Ax''|| = ||''x''|| for all ''x'' in ''V''. All isometries are [[injective]]. Isometries are [[morphism]]s between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with [[orthogonal matrix]]).
| |
| * Isometrical isomorphisms, i.e., ''A'' is an isometry which is [[surjective]] (and hence [[bijective]]). Isometrical isomorphisms are also known as unitary operators (compare with [[unitary matrix]]).
| |
| | |
| From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The [[spectral theorem]] provides a canonical form for symmetric, unitary and more generally [[normal operator]]s on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.
| |
| | |
| == Generalizations ==
| |
| Any of the axioms of an inner product may be weakened, yielding generalized notions. The generalizations that are closest to inner products occur where bilinearity and conjugate symmetry are retained, but positive-definiteness is weakened.
| |
| | |
| === Degenerate inner products ===
| |
| If ''V'' is a vector space and <math>\langle -, -\rangle</math> a semi-definite sesquilinear form,
| |
| then the function ‖''x''‖ = <math>\langle x, x\rangle^{1/2}</math> makes sense and satisfies all the properties of norm except that ‖''x''‖ = 0 does not imply ''x'' = 0 (such a functional is then called a [[semi-norm]]). We can produce an inner product space by considering the
| |
| quotient ''W'' = ''V''/{ ''x'' : ‖''x''‖ = 0}. The sesquilinear form <math>\langle -, -\rangle</math> factors through ''W''.
| |
| | |
| This construction is used in numerous contexts. The [[Gelfand–Naimark–Segal construction]] is a particularly important example of the use of this technique. Another example is the representation of [[Mercer's theorem|semi-definite kernel]]s on arbitrary sets.
| |
| | |
| === Nondegenerate conjugate symmetric forms ===
| |
| {{Main|Pseudo-Euclidean space}}
| |
| Alternatively, one may require that the pairing be a [[nondegenerate form]], meaning that for all non-zero ''x'' there exists some ''y'' such that <math>\langle x, y\rangle \neq 0,</math> though ''y'' need not equal ''x''; in other words, the induced map to the dual space <math>V \to V^*</math> is injective. This generalization is important in [[differential geometry]]: a manifold whose tangent spaces have an inner product is a [[Riemannian manifold]], while if this is related to nondegenerate conjugate symmetric form the manifold is a [[pseudo-Riemannian manifold]]. By [[Sylvester's law of inertia]], just as every inner product is similar to the dot product with positive weights on a set of vectors, every nondegenerate conjugate symmetric form is similar to the dot product with ''nonzero'' weights on a set of vectors, and the number of positive and negative weights are called respectively the positive index and negative index. Product of vectors in [[Minkowski space]] is an example of indefinite inner product, although, technically speaking, it is not an inner product according to the standard definition above. Minkowski space has four [[dimension (mathematics)|dimensions]] and indices 3 and 1 (assignment of [[sign (mathematics)|"+" and "−"]] to them [[sign convention#Metric signature|differs depending on conventions]]).
| |
| | |
| Purely algebraic statements (ones that do not use positivity) usually only rely on the nondegeneracy (the injective homomorphism <math>V \to V^*</math>) and thus hold more generally.
| |
| | |
| ==Related products==
| |
| The term "inner product" is opposed to [[outer product]], which is a slightly more general opposite. Simply, in coordinates, the inner product is the product of a 1 × ''n'' ''co''vector with an ''n'' × 1 vector, yielding a 1×1 matrix (a scalar), while the outer product is the product of an ''m''×1 vector with a 1×''n'' covector, yielding an ''m''×''n'' matrix. Note that the outer product is defined for different dimensions, while the inner product requires the same dimension. If the dimensions are the same, then the inner product is the ''trace'' of the outer product (trace only being properly defined for square matrices).
| |
| | |
| On an inner product space, or more generally a vector space with a [[nondegenerate form]] (so an isomorphism <math>V \to V^*</math>) vectors can be sent to covectors (in coordinates, via transpose), so one can take the inner product and outer product of two vectors, not simply of a vector and a covector.
| |
| | |
| In a quip: "inner is horizontal times vertical and shrinks down, outer is vertical times horizontal and expands out".
| |
| | |
| More abstractly, the outer product is the bilinear map <math>W \times V^* \to \operatorname{Hom}(V,W)</math> sending a vector and a covector to a rank 1 linear transformation ([[simple tensor]] of type (1,1)), while the inner product is the bilinear evaluation map <math>V^* \times V \to F</math> given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector/vector distinction.
| |
| | |
| The inner product and outer product should not be confused with the [[interior product]] and [[exterior product]], which are instead operations on [[vector field]]s and [[differential form]]s, or more generally on the [[exterior algebra]].
| |
| | |
| As a further complication, in [[geometric algebra]] the inner product and the ''exterior'' (Grassmann) product are combined in the geometric product (the Clifford product in a [[Clifford algebra]]) – the inner product sends two vectors (1-vectors) to a scalar (a 0-vector), while the exterior product sends two vectors to a bivector (2-vector) – and in this context the exterior product is usually called the "''outer'' (alternatively, wedge) product". The inner product is more correctly called a ''scalar'' product in this context, as the nondegenerate quadratic form in question need not be positive definite (need not be an inner product).
| |
| | |
| ==Notes and in-line references==
| |
| <references/>
| |
| | |
| ==See also==
| |
| * [[Bilinear form]]
| |
| * [[Dual space]]
| |
| * [[Dual pair]]
| |
| * [[Cross product]]
| |
| * [[Biorthogonal system]]
| |
| * [[Fubini–Study metric]]
| |
| * [[Energetic space]]
| |
| * [[Orthogonal group]]
| |
| * [[Space (mathematics)]]
| |
| * [[Normed vector space]]
| |
| | |
| == References ==
| |
| * {{Cite book | last1=Axler | first1=Sheldon | title=Linear Algebra Done Right | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-98258-8 | year=1997 | postscript=<!--None-->}}
| |
| * {{Cite book | last1=Emch | first1=Gerard G. | title=Algebraic methods in statistical mechanics and quantum field theory | publisher=[[Wiley-Interscience]] | isbn=978-0-471-23900-0 | year=1972 | postscript=<!--None-->}}
| |
| * {{Cite book | last1=Young | first1=Nicholas | title=An introduction to Hilbert space | publisher=[[Cambridge University Press]] | isbn=978-0-521-33717-5 | year=1988 | postscript=<!--None-->}}
| |
| | |
| <!-- AWB bots, please leave this space alone. -->
| |
| | |
| {{linear algebra}}
| |
| | |
| [[Category:Normed spaces]]
| |
| [[Category:Bilinear forms]]
| |