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Inner product space

From Wikipedia, the free encyclopedia
Vector space with generalized dot product
"Inner product" redirects here. For the inner product of coordinate vectors, seeDot product.
Geometric interpretation of the angle between two vectors defined using an inner product
Scalar product spaces, inner product spaces, Hermitian product spaces.
Scalar product spaces, over any field, have "scalar products" that are symmetrical and linear in the first argument. Hermitian product spaces are restricted to the field of complex numbers and have "Hermitian products" that are conjugate-symmetrical and linear in the first argument. Inner product spaces may be defined over any field, having "inner products" that are linear in the first argument, conjugate-symmetrical, and positive-definite. Unlike inner products, scalar products and Hermitian products need not be positive-definite.

Inmathematics, aninner product space[Note 1] is areal orcomplex vector space endowed with anoperation called aninner product. The inner product of two vectors in the space is ascalar, often denoted withangle brackets such as in⟨a,b⟩{\displaystyle \langle a,b\rangle }. Inner products allow formal definitions of intuitive geometric notions, such as lengths,angles, andorthogonality (zero inner product) of vectors. Inner product spaces generalizeEuclidean vector spaces, in which the inner product is thedot product orscalar product ofCartesian coordinates. Inner product spaces of infinitedimensions are widely used infunctional analysis. Inner product spaces over thefield ofcomplex numbers are sometimes referred to asunitary spaces. The first usage of the concept of a vector space with an inner product is due toGiuseppe Peano, in 1898.[3]

An inner product naturally induces an associatednorm, (denoted|x|{\displaystyle |x|} and|y|{\displaystyle |y|} in the picture); so, every inner product space is anormed vector space. If this normed space is alsocomplete (that is, aBanach space) then the inner product space is aHilbert space.[1] If an inner product spaceH is not a Hilbert space, it can beextended bycompletion to a Hilbert spaceH¯.{\displaystyle {\overline {H}}.} This means thatH{\displaystyle H} is alinear subspace ofH¯,{\displaystyle {\overline {H}},} the inner product ofH{\displaystyle H} is therestriction of that ofH¯,{\displaystyle {\overline {H}},} andH{\displaystyle H} isdense inH¯{\displaystyle {\overline {H}}} for thetopology defined by the norm.[1][4]

Definition

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In this article,F denotes afield that is either thereal numbersR,{\displaystyle \mathbb {R} ,} or thecomplex numbersC.{\displaystyle \mathbb {C} .} Ascalar is thus an element ofF. A bar over an expression representing a scalar denotes thecomplex conjugate of this scalar. A zero vector is denoted0{\displaystyle \mathbf {0} } for distinguishing it from the scalar0.

Aninner product space is avector spaceV over the fieldF together with aninner product, that is, a map⟨⋅,⋅⟩:V×V→F{\displaystyle \langle \cdot \operatorname {,} \cdot \rangle :V\times V\to F}that satisfies the following three properties for all vectorsx,y,z∈V{\displaystyle x,y,z\in V} and all scalarsa,b∈F{\displaystyle a,b\in F}.[5][6]

If the positive-definiteness condition is replaced by merely requiring that⟨x,x⟩≥0{\displaystyle \langle x,x\rangle \geq 0} for allx{\displaystyle x}, then one obtains the definition ofpositive semi-definite Hermitian form. A positive semi-definite Hermitian form⟨⋅,⋅⟩{\displaystyle \langle \cdot ,\cdot \rangle } is an inner product if and only if for allx{\displaystyle x}, if⟨x,x⟩=0{\displaystyle \langle x,x\rangle =0} thenx=0{\displaystyle x=\mathbf {0} }.[7]

Basic properties

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In the following properties, which result almost immediately from the definition of an inner product,x,y andz are arbitrary vectors, anda andb are arbitrary scalars.

OverR{\displaystyle \mathbb {R} }, conjugate-symmetry reduces to symmetry, and sesquilinearity reduces tobilinearity. Hence an inner product on a real vector space is apositive-definite symmetricbilinear form. Thebinomial expansion of a square becomes⟨x+y,x+y⟩=⟨x,x⟩+2⟨x,y⟩+⟨y,y⟩.{\displaystyle \langle x+y,x+y\rangle =\langle x,x\rangle +2\langle x,y\rangle +\langle y,y\rangle .}

Notation

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Several notations are used for inner products, including⟨⋅,⋅⟩{\displaystyle \langle \cdot ,\cdot \rangle },(⋅,⋅){\displaystyle \left(\cdot ,\cdot \right)},⟨⋅|⋅⟩{\displaystyle \langle \cdot |\cdot \rangle } and(⋅|⋅){\displaystyle \left(\cdot |\cdot \right)}, as well as the usual dot product.

Convention variant

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Some authors, especially inphysics andmatrix algebra, prefer to define inner products and sesquilinear forms with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second.Bra–ket notation inquantum mechanics also uses slightly different notation, i.e.⟨⋅|⋅⟩{\displaystyle \langle \cdot |\cdot \rangle }, where⟨x|y⟩:=(y,x){\displaystyle \langle x|y\rangle :=\left(y,x\right)}.

Examples

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Real and complex numbers

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Among the simplest examples of inner product spaces areR{\displaystyle \mathbb {R} } andC.{\displaystyle \mathbb {C} .} Thereal numbersR{\displaystyle \mathbb {R} } are a vector space overR{\displaystyle \mathbb {R} } that becomes an inner product space with arithmetic multiplication as its inner product:⟨x,y⟩:=xy for x,y∈R.{\displaystyle \langle x,y\rangle :=xy\quad {\text{ for }}x,y\in \mathbb {R} .}

Thecomplex numbersC{\displaystyle \mathbb {C} } are a vector space overC{\displaystyle \mathbb {C} } that becomes an inner product space with the inner product⟨x,y⟩:=xy¯ for x,y∈C.{\displaystyle \langle x,y\rangle :=x{\overline {y}}\quad {\text{ for }}x,y\in \mathbb {C} .} Unlike with the real numbers, the assignment(x,y)↦xy{\displaystyle (x,y)\mapsto xy} doesnot define a complex inner product onC.{\displaystyle \mathbb {C} .}

Euclidean vector space

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More generally, therealn{\displaystyle n}-spaceRn{\displaystyle \mathbb {R} ^{n}} with thedot product is an inner product space, an example of aEuclidean vector space.⟨[x1⋮xn],[y1⋮yn]⟩=xTy=∑i=1nxiyi=x1y1+⋯+xnyn,{\displaystyle \left\langle {\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}},{\begin{bmatrix}y_{1}\\\vdots \\y_{n}\end{bmatrix}}\right\rangle =x^{\operatorname {T} }y=\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+\cdots +x_{n}y_{n},}wherexT{\displaystyle x^{\operatorname {T} }} is thetranspose ofx.{\displaystyle x.}

A function⟨⋅,⋅⟩:Rn×Rn→R{\displaystyle \langle \,\cdot ,\cdot \,\rangle :\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} } is an inner product onRn{\displaystyle \mathbb {R} ^{n}} if and only if there exists asymmetricpositive-definite matrixM{\displaystyle \mathbf {M} } such that⟨x,y⟩=xTMy{\displaystyle \langle x,y\rangle =x^{\operatorname {T} }\mathbf {M} y} for allx,y∈Rn.{\displaystyle x,y\in \mathbb {R} ^{n}.} IfM{\displaystyle \mathbf {M} } is theidentity matrix then⟨x,y⟩=xTMy{\displaystyle \langle x,y\rangle =x^{\operatorname {T} }\mathbf {M} y} is the dot product. For another example, ifn=2{\displaystyle n=2} andM=[abbd]{\displaystyle \mathbf {M} ={\begin{bmatrix}a&b\\b&d\end{bmatrix}}} is positive-definite (which happens if and only ifdetM=ad−b2>0{\displaystyle \det \mathbf {M} =ad-b^{2}>0} and one/both diagonal elements are positive) then for anyx:=[x1,x2]T,y:=[y1,y2]T∈R2,{\displaystyle x:=\left[x_{1},x_{2}\right]^{\operatorname {T} },y:=\left[y_{1},y_{2}\right]^{\operatorname {T} }\in \mathbb {R} ^{2},}⟨x,y⟩:=xTMy=[x1,x2][abbd][y1y2]=ax1y1+bx1y2+bx2y1+dx2y2.{\displaystyle \langle x,y\rangle :=x^{\operatorname {T} }\mathbf {M} y=\left[x_{1},x_{2}\right]{\begin{bmatrix}a&b\\b&d\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}=ax_{1}y_{1}+bx_{1}y_{2}+bx_{2}y_{1}+dx_{2}y_{2}.} As mentioned earlier, every inner product onR2{\displaystyle \mathbb {R} ^{2}} is of this form (whereb∈R,a>0{\displaystyle b\in \mathbb {R} ,a>0} andd>0{\displaystyle d>0} satisfyad>b2{\displaystyle ad>b^{2}}).

Complex coordinate space

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The general form of an inner product onCn{\displaystyle \mathbb {C} ^{n}} is known as theHermitian form and is given by⟨x,y⟩=y†Mx=x†My¯,{\displaystyle \langle x,y\rangle =y^{\dagger }\mathbf {M} x={\overline {x^{\dagger }\mathbf {M} y}},}whereM{\displaystyle M} is anyHermitianpositive-definite matrix andy†{\displaystyle y^{\dagger }} is theconjugate transpose ofy.{\displaystyle y.} For the real case, this corresponds to the dot product of the results of directionally-differentscaling of the two vectors, with positivescale factors and orthogonal directions of scaling. It is aweighted-sum version of the dot product with positive weights—up to an orthogonal transformation.

Hilbert space

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The article onHilbert spaces has several examples of inner product spaces, wherein the metric induced by the inner product yields acomplete metric space. An example of an inner product space which induces an incomplete metric is the spaceC([a,b]){\displaystyle C([a,b])} of continuous complex valued functionsf{\displaystyle f} andg{\displaystyle g} on the interval[a,b].{\displaystyle [a,b].} The inner product is⟨f,g⟩=∫abf(t)g(t)¯dt.{\displaystyle \langle f,g\rangle =\int _{a}^{b}f(t){\overline {g(t)}}\,\mathrm {d} t.}This space is not complete; consider for example, for the interval[−1, 1] the sequence of continuous "step" functions,{fk}k,{\displaystyle \{f_{k}\}_{k},} defined by:fk(t)={0t∈[−1,0]1t∈[1k,1]ktt∈(0,1k){\displaystyle f_{k}(t)={\begin{cases}0&t\in [-1,0]\\1&t\in \left[{\tfrac {1}{k}},1\right]\\kt&t\in \left(0,{\tfrac {1}{k}}\right)\end{cases}}}

This sequence is aCauchy sequence for the norm induced by the preceding inner product, which does not converge to acontinuous function.

Random variables

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For realrandom variablesX{\displaystyle X} andY,{\displaystyle Y,} theexpected value of their product⟨X,Y⟩=E[XY]{\displaystyle \langle X,Y\rangle =\mathbb {E} [XY]}is an inner product.[8][9][10] In this case,⟨X,X⟩=0{\displaystyle \langle X,X\rangle =0} if and only ifP[X=0]=1{\displaystyle \mathbb {P} [X=0]=1} (that is,X=0{\displaystyle X=0}almost surely), whereP{\displaystyle \mathbb {P} } denotes theprobability of the event. This definition of expectation as inner product can be extended torandom vectors as well.

Complex matrices

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The inner product for complex square matrices of the same size is theFrobenius inner product⟨A,B⟩:=tr⁡(AB†){\displaystyle \langle A,B\rangle :=\operatorname {tr} \left(AB^{\dagger }\right)}. Since trace and transposition are linear and the conjugation is on the second matrix, it is a sesquilinear operator. We further get Hermitian symmetry by,⟨A,B⟩=tr⁡(AB†)=tr⁡(BA†)¯=⟨B,A⟩¯{\displaystyle \langle A,B\rangle =\operatorname {tr} \left(AB^{\dagger }\right)={\overline {\operatorname {tr} \left(BA^{\dagger }\right)}}={\overline {\left\langle B,A\right\rangle }}}Finally, since forA{\displaystyle A} nonzero,⟨A,A⟩=∑ij|Aij|2>0{\displaystyle \langle A,A\rangle =\sum _{ij}\left|A_{ij}\right|^{2}>0}, we get that the Frobenius inner product is positive definite too, and so is an inner product.

Vector spaces with forms

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On an inner product space, or more generally a vector space with anondegenerate form (hence an isomorphismV→V∗{\displaystyle V\to V^{*}}), vectors can be sent to covectors (in coordinates, via transpose), so that one can take the inner product and outer product of two vectors—not simply of a vector and a covector.

Basic results, terminology, and definitions

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Norm properties

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Every inner product space induces anorm, called itscanonical norm, that is defined by‖x‖=⟨x,x⟩.{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}.} With this norm, every inner product space becomes anormed vector space.

So, every general property of normed vector spaces applies to inner product spaces. In particular, one has the following properties:

Absolute homogeneity
‖ax‖=|a|‖x‖{\displaystyle \|ax\|=|a|\,\|x\|}for everyx∈V{\displaystyle x\in V} anda∈F{\displaystyle a\in F}(this results from⟨ax,ax⟩=aa¯⟨x,x⟩{\displaystyle \langle ax,ax\rangle =a{\overline {a}}\langle x,x\rangle }).
Triangle inequality
‖x+y‖≤‖x‖+‖y‖{\displaystyle \|x+y\|\leq \|x\|+\|y\|}forx,y∈V.{\displaystyle x,y\in V.}These two properties show that one has indeed a norm.
Cauchy–Schwarz inequality
|⟨x,y⟩|≤‖x‖‖y‖{\displaystyle |\langle x,y\rangle |\leq \|x\|\,\|y\|}for everyx,y∈V,{\displaystyle x,y\in V,}with equality if and only ifx{\displaystyle x} andy{\displaystyle y} arelinearly dependent.
Parallelogram law
‖x+y‖2+‖x−y‖2=2‖x‖2+2‖y‖2{\displaystyle \|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}}for everyx,y∈V.{\displaystyle x,y\in V.}The parallelogram law is a necessary and sufficient condition for a norm to be defined by an inner product.
Polarization identity
‖x+y‖2=‖x‖2+‖y‖2+2Re⁡⟨x,y⟩{\displaystyle \|x+y\|^{2}=\|x\|^{2}+\|y\|^{2}+2\operatorname {Re} \langle x,y\rangle }for everyx,y∈V.{\displaystyle x,y\in V.}The inner product can be retrieved from the norm by the polarization identity, since its imaginary part is the real part of⟨x,iy⟩.{\displaystyle \langle x,iy\rangle .}
Ptolemy's inequality
‖x−y‖‖z‖ + ‖y−z‖‖x‖ ≥ ‖x−z‖‖y‖{\displaystyle \|x-y\|\,\|z\|~+~\|y-z\|\,\|x\|~\geq ~\|x-z\|\,\|y\|}for everyx,y,z∈V.{\displaystyle x,y,z\in V.}Ptolemy's inequality is a necessary and sufficient condition for aseminorm to be the norm defined by an inner product.[11]

Orthogonality

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Orthogonality
Two vectorsx{\displaystyle x} andy{\displaystyle y} are said to beorthogonal, often writtenx⊥y,{\displaystyle x\perp y,} if their inner product is zero, that is, if⟨x,y⟩=0.{\displaystyle \langle x,y\rangle =0.}
This happens if and only if‖x‖≤‖x+sy‖{\displaystyle \|x\|\leq \|x+sy\|} for all scalarss,{\displaystyle s,}[12] and if and only if the real-valued functionf(s):=‖x+sy‖2−‖x‖2{\displaystyle f(s):=\|x+sy\|^{2}-\|x\|^{2}} is non-negative. (This is a consequence of the fact that, ify≠0{\displaystyle y\neq 0} then the scalars0=−⟨x,y⟩¯‖y‖2{\displaystyle s_{0}=-{\tfrac {\overline {\langle x,y\rangle }}{\|y\|^{2}}}} minimizesf{\displaystyle f} with valuef(s0)=−|⟨x,y⟩|2‖y‖2,{\displaystyle f\left(s_{0}\right)=-{\tfrac {|\langle x,y\rangle |^{2}}{\|y\|^{2}}},} which is always non positive).
For acomplex inner product spaceH,{\displaystyle H,} a linear operatorT:V→V{\displaystyle T:V\to V} is identically0{\displaystyle 0} if and only ifx⊥Tx{\displaystyle x\perp Tx} for everyx∈V.{\displaystyle x\in V.}[12] This is not true in general for real inner product spaces, as it is a consequence of conjugate symmetry being distinct from symmetry for complex inner products. A counterexample in a real inner product space isT{\displaystyle T} a 90° rotation inR2{\displaystyle \mathbb {R} ^{2}}, which maps every vector to an orthogonal vector but is not identically0{\displaystyle 0}.
Orthogonal complement
Theorthogonal complement of a subsetC⊆V{\displaystyle C\subseteq V} is the setC⊥{\displaystyle C^{\bot }} of the vectors that are orthogonal to all elements ofC; that is,C⊥:={y∈V:⟨y,c⟩=0 for all c∈C}.{\displaystyle C^{\bot }:=\{\,y\in V:\langle y,c\rangle =0{\text{ for all }}c\in C\,\}.} This setC⊥{\displaystyle C^{\bot }} is always a closed vector subspace ofV{\displaystyle V} and if theclosureclV⁡C{\displaystyle \operatorname {cl} _{V}C} ofC{\displaystyle C} inV{\displaystyle V} is a vector subspace thenclV⁡C=(C⊥)⊥.{\displaystyle \operatorname {cl} _{V}C=\left(C^{\bot }\right)^{\bot }.}
Pythagorean theorem
Ifx{\displaystyle x} andy{\displaystyle y} are orthogonal, then‖x‖2+‖y‖2=‖x+y‖2.{\displaystyle \|x\|^{2}+\|y\|^{2}=\|x+y\|^{2}.}This may be proved by expressing the squared norms in terms of the inner products, using additivity for expanding the right-hand side of the equation.
The namePythagorean theorem arises from the geometric interpretation inEuclidean geometry.
Parseval's identity
Aninduction on the Pythagorean theorem yields: ifx1,…,xn{\displaystyle x_{1},\ldots ,x_{n}} are pairwise orthogonal, then∑i=1n‖xi‖2=‖∑i=1nxi‖2.{\displaystyle \sum _{i=1}^{n}\|x_{i}\|^{2}=\left\|\sum _{i=1}^{n}x_{i}\right\|^{2}.}
Angle
When⟨x,y⟩{\displaystyle \langle x,y\rangle } is a real number then the Cauchy–Schwarz inequality implies that⟨x,y⟩‖x‖‖y‖∈[−1,1],{\textstyle {\frac {\langle x,y\rangle }{\|x\|\,\|y\|}}\in [-1,1],} and thus that∠(x,y)=arccos⁡⟨x,y⟩‖x‖‖y‖,{\displaystyle \angle (x,y)=\arccos {\frac {\langle x,y\rangle }{\|x\|\,\|y\|}},}is a real number. This allows defining the (non oriented)angle of two vectors in modern definitions ofEuclidean geometry in terms oflinear algebra. This is also used indata analysis, under the name "cosine similarity", for comparing two vectors of data.Furthermore, if⟨x,y⟩{\displaystyle \langle x,y\rangle } is negative, the angle∠(x,y){\displaystyle \angle (x,y)} is larger than 90 degrees. This property is often used in computer graphics (e.g., inback-face culling) to analyze a direction without having to evaluatetrigonometric functions.

Real and complex parts of inner products

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Suppose that⟨⋅,⋅⟩{\displaystyle \langle \cdot ,\cdot \rangle } is an inner product onV{\displaystyle V} (so it is antilinear in its second argument). Thepolarization identity shows that thereal part of the inner product isRe⁡⟨x,y⟩=14(‖x+y‖2−‖x−y‖2).{\displaystyle \operatorname {Re} \langle x,y\rangle ={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}\right).}

IfV{\displaystyle V} is a real vector space then⟨x,y⟩=Re⁡⟨x,y⟩=14(‖x+y‖2−‖x−y‖2){\displaystyle \langle x,y\rangle =\operatorname {Re} \langle x,y\rangle ={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}\right)}and theimaginary part (also called thecomplex part) of⟨⋅,⋅⟩{\displaystyle \langle \cdot ,\cdot \rangle } is always0.{\displaystyle 0.}

Assume for the rest of this section thatV{\displaystyle V} is a complex vector space.Thepolarization identity for complex vector spaces shows that⟨x, y⟩=14(‖x+y‖2−‖x−y‖2+i‖x+iy‖2−i‖x−iy‖2)=Re⁡⟨x,y⟩+iRe⁡⟨x,iy⟩.{\displaystyle {\begin{alignedat}{4}\langle x,\ y\rangle &={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}+i\|x+iy\|^{2}-i\|x-iy\|^{2}\right)\\&=\operatorname {Re} \langle x,y\rangle +i\operatorname {Re} \langle x,iy\rangle .\\\end{alignedat}}}

The map defined by⟨x∣y⟩=⟨y,x⟩{\displaystyle \langle x\mid y\rangle =\langle y,x\rangle } for allx,y∈V{\displaystyle x,y\in V} satisfies the axioms of the inner product except that it is antilinear in itsfirst, rather than its second, argument. The real part of both⟨x∣y⟩{\displaystyle \langle x\mid y\rangle } and⟨x,y⟩{\displaystyle \langle x,y\rangle } are equal toRe⁡⟨x,y⟩{\displaystyle \operatorname {Re} \langle x,y\rangle } but the inner products differ in their complex part:⟨x∣y⟩=14(‖x+y‖2−‖x−y‖2−i‖x+iy‖2+i‖x−iy‖2)=Re⁡⟨x,y⟩−iRe⁡⟨x,iy⟩.{\displaystyle {\begin{alignedat}{4}\langle x\mid y\rangle &={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}-i\|x+iy\|^{2}+i\|x-iy\|^{2}\right)\\&=\operatorname {Re} \langle x,y\rangle -i\operatorname {Re} \langle x,iy\rangle .\\\end{alignedat}}}

The last equality is similar to the formulaexpressing a linear functional in terms of its real part.

These formulas show that every complex inner product is completely determined by its real part. Moreover, this real part defines an inner product onV,{\displaystyle V,} considered as a real vector space. There is thus a one-to-one correspondence between complex inner products on a complex vector spaceV,{\displaystyle V,} and real inner products onV.{\displaystyle V.}

For example, suppose thatV=Cn{\displaystyle V=\mathbb {C} ^{n}} for some integern>0.{\displaystyle n>0.} WhenV{\displaystyle V} is considered as a real vector space in the usual way (meaning that it is identified with the2n−{\displaystyle 2n-}dimensional real vector spaceR2n,{\displaystyle \mathbb {R} ^{2n},} with each(a1+ib1,…,an+ibn)∈Cn{\displaystyle \left(a_{1}+ib_{1},\ldots ,a_{n}+ib_{n}\right)\in \mathbb {C} ^{n}} identified with(a1,b1,…,an,bn)∈R2n{\displaystyle \left(a_{1},b_{1},\ldots ,a_{n},b_{n}\right)\in \mathbb {R} ^{2n}}), then thedot productx⋅y=(x1,…,x2n)⋅(y1,…,y2n):=x1y1+⋯+x2ny2n{\displaystyle x\,\cdot \,y=\left(x_{1},\ldots ,x_{2n}\right)\,\cdot \,\left(y_{1},\ldots ,y_{2n}\right):=x_{1}y_{1}+\cdots +x_{2n}y_{2n}} defines a real inner product on this space. The unique complex inner product⟨⋅,⋅⟩{\displaystyle \langle \,\cdot ,\cdot \,\rangle } onV=Cn{\displaystyle V=\mathbb {C} ^{n}} induced by the dot product is the map that sendsc=(c1,…,cn),d=(d1,…,dn)∈Cn{\displaystyle c=\left(c_{1},\ldots ,c_{n}\right),d=\left(d_{1},\ldots ,d_{n}\right)\in \mathbb {C} ^{n}} to⟨c,d⟩:=c1d1¯+⋯+cndn¯{\displaystyle \langle c,d\rangle :=c_{1}{\overline {d_{1}}}+\cdots +c_{n}{\overline {d_{n}}}} (because the real part of this map⟨⋅,⋅⟩{\displaystyle \langle \,\cdot ,\cdot \,\rangle } is equal to the dot product).

Real vs. complex inner products

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LetVR{\displaystyle V_{\mathbb {R} }} denoteV{\displaystyle V} considered as a vector space over the real numbers rather than complex numbers.Thereal part of the complex inner product⟨x,y⟩{\displaystyle \langle x,y\rangle } is the map⟨x,y⟩R=Re⁡⟨x,y⟩ : VR×VR→R,{\displaystyle \langle x,y\rangle _{\mathbb {R} }=\operatorname {Re} \langle x,y\rangle ~:~V_{\mathbb {R} }\times V_{\mathbb {R} }\to \mathbb {R} ,} which necessarily forms a real inner product on the real vector spaceVR.{\displaystyle V_{\mathbb {R} }.} Every inner product on a real vector space is abilinear andsymmetric map.

For example, ifV=C{\displaystyle V=\mathbb {C} } with inner product⟨x,y⟩=xy¯,{\displaystyle \langle x,y\rangle =x{\overline {y}},} whereV{\displaystyle V} is a vector space over the fieldC,{\displaystyle \mathbb {C} ,} thenVR=R2{\displaystyle V_{\mathbb {R} }=\mathbb {R} ^{2}} is a vector space overR{\displaystyle \mathbb {R} } and⟨x,y⟩R{\displaystyle \langle x,y\rangle _{\mathbb {R} }} is thedot productx⋅y,{\displaystyle x\cdot y,} wherex=a+ib∈V=C{\displaystyle x=a+ib\in V=\mathbb {C} } is identified with the point(a,b)∈VR=R2{\displaystyle (a,b)\in V_{\mathbb {R} }=\mathbb {R} ^{2}} (and similarly fory{\displaystyle y}); thus the standard inner product⟨x,y⟩=xy¯,{\displaystyle \langle x,y\rangle =x{\overline {y}},} onC{\displaystyle \mathbb {C} } is an "extension" the dot product . Also, had⟨x,y⟩{\displaystyle \langle x,y\rangle } been instead defined to be thesymmetric map⟨x,y⟩=xy{\displaystyle \langle x,y\rangle =xy} (rather than the usualconjugate symmetric map⟨x,y⟩=xy¯{\displaystyle \langle x,y\rangle =x{\overline {y}}}) then its real part⟨x,y⟩R{\displaystyle \langle x,y\rangle _{\mathbb {R} }} wouldnot be the dot product; furthermore, without the complex conjugate, ifx∈C{\displaystyle x\in \mathbb {C} } butx∉R{\displaystyle x\not \in \mathbb {R} } then⟨x,x⟩=xx=x2∉[0,∞){\displaystyle \langle x,x\rangle =xx=x^{2}\not \in [0,\infty )} so the assignmentx↦⟨x,x⟩{\textstyle x\mapsto {\sqrt {\langle x,x\rangle }}} would not define a norm.

The next examples show that although real and complex inner products have many properties and results in common, they are not entirely interchangeable.For instance, if⟨x,y⟩=0{\displaystyle \langle x,y\rangle =0} then⟨x,y⟩R=0,{\displaystyle \langle x,y\rangle _{\mathbb {R} }=0,} but the next example shows that the converse is in generalnot true.Given anyx∈V,{\displaystyle x\in V,} the vectorix{\displaystyle ix} (which is the vectorx{\displaystyle x} rotated by 90°) belongs toV{\displaystyle V} and so also belongs toVR{\displaystyle V_{\mathbb {R} }} (although scalar multiplication ofx{\displaystyle x} byi=−1{\displaystyle i={\sqrt {-1}}} is not defined inVR,{\displaystyle V_{\mathbb {R} },} the vector inV{\displaystyle V} denoted byix{\displaystyle ix} is nevertheless still also an element ofVR{\displaystyle V_{\mathbb {R} }}). For the complex inner product,⟨x,ix⟩=−i‖x‖2,{\displaystyle \langle x,ix\rangle =-i\|x\|^{2},} whereas for the real inner product the value is always⟨x,ix⟩R=0.{\displaystyle \langle x,ix\rangle _{\mathbb {R} }=0.}

If⟨⋅,⋅⟩{\displaystyle \langle \,\cdot ,\cdot \,\rangle } is a complex inner product andA:V→V{\displaystyle A:V\to V} is a continuous linear operator that satisfies⟨x,Ax⟩=0{\displaystyle \langle x,Ax\rangle =0} for allx∈V,{\displaystyle x\in V,} thenA=0.{\displaystyle A=0.} This statement is no longer true if⟨⋅,⋅⟩{\displaystyle \langle \,\cdot ,\cdot \,\rangle } is instead a real inner product, as this next example shows. Suppose thatV=C{\displaystyle V=\mathbb {C} } has the inner product⟨x,y⟩:=xy¯{\displaystyle \langle x,y\rangle :=x{\overline {y}}} mentioned above. Then the mapA:V→V{\displaystyle A:V\to V} defined byAx=ix{\displaystyle Ax=ix} is a linear map (linear for bothV{\displaystyle V} andVR{\displaystyle V_{\mathbb {R} }}) that denotes rotation by90∘{\displaystyle 90^{\circ }} in the plane. Becausex{\displaystyle x} andAx{\displaystyle Ax} are perpendicular vectors and⟨x,Ax⟩R{\displaystyle \langle x,Ax\rangle _{\mathbb {R} }} is just the dot product,⟨x,Ax⟩R=0{\displaystyle \langle x,Ax\rangle _{\mathbb {R} }=0} for all vectorsx;{\displaystyle x;} nevertheless, this rotation mapA{\displaystyle A} is certainly not identically0.{\displaystyle 0.} In contrast, using the complex inner product gives⟨x,Ax⟩=−i‖x‖2,{\displaystyle \langle x,Ax\rangle =-i\|x\|^{2},} which (as expected) is not identically zero.

Orthonormal sequences

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See also:Orthogonal basis andOrthonormal basis

LetV{\displaystyle V} be a finite dimensional inner product space of dimensionn.{\displaystyle n.} Recall that everybasis ofV{\displaystyle V} consists of exactlyn{\displaystyle n} linearly independent vectors. Using theGram–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{e1,…,en}{\displaystyle \{e_{1},\ldots ,e_{n}\}} is orthonormal if⟨ei,ej⟩=0{\displaystyle \langle e_{i},e_{j}\rangle =0} for everyi≠j{\displaystyle i\neq j} and⟨ei,ei⟩=‖ea‖2=1{\displaystyle \langle e_{i},e_{i}\rangle =\|e_{a}\|^{2}=1} for each indexi.{\displaystyle i.}

This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way. LetV{\displaystyle V} be any inner product space. Then a collectionE={ea}a∈A{\displaystyle E=\left\{e_{a}\right\}_{a\in A}}is abasis forV{\displaystyle V} if the subspace ofV{\displaystyle V} generated by finite linear combinations of elements ofE{\displaystyle E} is dense inV{\displaystyle V} (in the norm induced by the inner product). Say thatE{\displaystyle E} is anorthonormal basis forV{\displaystyle V} if it is a basis and⟨ea,eb⟩=0{\displaystyle \left\langle e_{a},e_{b}\right\rangle =0}ifa≠b{\displaystyle a\neq b} and⟨ea,ea⟩=‖ea‖2=1{\displaystyle \langle e_{a},e_{a}\rangle =\|e_{a}\|^{2}=1} for alla,b∈A.{\displaystyle a,b\in A.}

Using an infinite-dimensional analog of the Gram-Schmidt process one may show:

Theorem. Anyseparable inner product space has an orthonormal basis.

Using theHausdorff maximal principle and the fact that in acomplete inner product space orthogonal projection onto linear subspaces is well-defined, one may also show that

Theorem. Anycomplete inner product space 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'sA Hilbert Space Problem Book (see the references).[citation needed]

Proof
Recall that the dimension of an inner product space is thecardinality of a maximal orthonormal system that it contains (byZorn's lemma it contains at least one, and any two have the same cardinality). An orthonormal basis is certainly a maximal orthonormal system but the converse need not hold in general. IfG{\displaystyle G} is a dense subspace of an inner product spaceV,{\displaystyle V,} then any orthonormal basis forG{\displaystyle G} is automatically an orthonormal basis forV.{\displaystyle V.} Thus, it suffices to construct an inner product spaceV{\displaystyle V} with a dense subspaceG{\displaystyle G} whose dimension is strictly smaller than that ofV.{\displaystyle V.}

LetK{\displaystyle K} be aHilbert space of dimensionℵ0.{\displaystyle \aleph _{0}.} (for instance,K=ℓ2(N){\displaystyle K=\ell ^{2}(\mathbb {N} )}). LetE{\displaystyle E} be an orthonormal basis ofK,{\displaystyle K,} so|E|=ℵ0.{\displaystyle |E|=\aleph _{0}.} ExtendE{\displaystyle E} to aHamel basisE∪F{\displaystyle E\cup F} forK,{\displaystyle K,}whereE∩F=∅.{\displaystyle E\cap F=\varnothing .} Since it is known that theHamel dimension ofK{\displaystyle K} isc,{\displaystyle c,} the cardinality of the continuum, it must be that|F|=c.{\displaystyle |F|=c.}

LetL{\displaystyle L} be a Hilbert space of dimensionc{\displaystyle c} (for instance,L=ℓ2(R){\displaystyle L=\ell ^{2}(\mathbb {R} )}). LetB{\displaystyle B} be an orthonormal basis forL{\displaystyle L} and letφ:F→B{\displaystyle \varphi :F\to B} be a bijection. Then there is a linear transformationT:K→L{\displaystyle T:K\to L} such thatTf=φ(f){\displaystyle Tf=\varphi (f)} forf∈F,{\displaystyle f\in F,} andTe=0{\displaystyle Te=0} fore∈E.{\displaystyle e\in E.}

LetV=K⊕L{\displaystyle V=K\oplus L} and letG={(k,Tk):k∈K}{\displaystyle G=\{(k,Tk):k\in K\}} be the graph ofT.{\displaystyle T.} LetG¯{\displaystyle {\overline {G}}} be the closure ofG{\displaystyle G} inV{\displaystyle V}; we will showG¯=V.{\displaystyle {\overline {G}}=V.} Since for anye∈E{\displaystyle e\in E} we have(e,0)∈G,{\displaystyle (e,0)\in G,} it follows thatK⊕0⊆G¯.{\displaystyle K\oplus 0\subseteq {\overline {G}}.}

Next, ifb∈B,{\displaystyle b\in B,} thenb=Tf{\displaystyle b=Tf} for somef∈F⊆K,{\displaystyle f\in F\subseteq K,} so(f,b)∈G⊆G¯{\displaystyle (f,b)\in G\subseteq {\overline {G}}}; since(f,0)∈G¯{\displaystyle (f,0)\in {\overline {G}}} as well, we also have(0,b)∈G¯.{\displaystyle (0,b)\in {\overline {G}}.} It follows that0⊕L⊆G¯,{\displaystyle 0\oplus L\subseteq {\overline {G}},} soG¯=V,{\displaystyle {\overline {G}}=V,} andG{\displaystyle G} is dense inV.{\displaystyle V.}

Finally,{(e,0):e∈E}{\displaystyle \{(e,0):e\in E\}} is a maximal orthonormal set inG{\displaystyle G}; if0=⟨(e,0),(k,Tk)⟩=⟨e,k⟩+⟨0,Tk⟩=⟨e,k⟩{\displaystyle 0=\langle (e,0),(k,Tk)\rangle =\langle e,k\rangle +\langle 0,Tk\rangle =\langle e,k\rangle }for alle∈E{\displaystyle e\in E} thenk=0,{\displaystyle k=0,} so(k,Tk)=(0,0){\displaystyle (k,Tk)=(0,0)} is the zero vector inG.{\displaystyle G.} Hence the dimension ofG{\displaystyle G} is|E|=ℵ0,{\displaystyle |E|=\aleph _{0},} whereas it is clear that the dimension ofV{\displaystyle V} isc.{\displaystyle c.} This completes the proof.

Parseval's identity leads immediately to the following theorem:

Theorem. LetV{\displaystyle V} be a separable inner product space and{ek}k{\displaystyle \left\{e_{k}\right\}_{k}} an orthonormal basis ofV.{\displaystyle V.} Then the mapx↦{⟨ek,x⟩}k∈N{\displaystyle x\mapsto {\bigl \{}\langle e_{k},x\rangle {\bigr \}}_{k\in \mathbb {N} }}is an isometric linear mapV→ℓ2{\displaystyle V\rightarrow \ell ^{2}} with a dense image.

This theorem can be regarded as an abstract form ofFourier series, in which an arbitrary orthonormal basis plays the role of the sequence oftrigonometric polynomials. Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, providedℓ2{\displaystyle \ell ^{2}} is defined appropriately, as is explained in the articleHilbert space). In particular, we obtain the following result in the theory of Fourier series:

Theorem. LetV{\displaystyle V} be the inner product spaceC[−π,π].{\displaystyle C[-\pi ,\pi ].} Then the sequence (indexed on set of all integers) of continuous functionsek(t)=eikt2π{\displaystyle e_{k}(t)={\frac {e^{ikt}}{\sqrt {2\pi }}}}is an orthonormal basis of the spaceC[−π,π]{\displaystyle C[-\pi ,\pi ]} with theL2{\displaystyle L^{2}} inner product. The mappingf↦12π{∫−ππf(t)e−iktdt}k∈Z{\displaystyle f\mapsto {\frac {1}{\sqrt {2\pi }}}\left\{\int _{-\pi }^{\pi }f(t)e^{-ikt}\,\mathrm {d} t\right\}_{k\in \mathbb {Z} }}is an isometric linear map with dense image.

Orthogonality of the sequence{ek}k{\displaystyle \{e_{k}\}_{k}} follows immediately from the fact that ifk≠j,{\displaystyle k\neq j,} then∫−ππe−i(j−k)tdt=0.{\displaystyle \int _{-\pi }^{\pi }e^{-i(j-k)t}\,\mathrm {d} t=0.}

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 theinner product norm, follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous periodic functions on[−π,π]{\displaystyle [-\pi ,\pi ]} with the uniform norm. This is the content of theWeierstrass theorem on the uniform density of trigonometric polynomials.

Operators on inner product spaces

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Main article:Operator theory

Several types oflinear mapsA:V→W{\displaystyle A:V\to W} between inner product spacesV{\displaystyle V} andW{\displaystyle W} are of relevance:

From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. Thespectral theorem provides a canonical form for symmetric, unitary and more generallynormal operators on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.[13]

Generalizations

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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

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Main article:Krein space

IfV{\displaystyle V} is a vector space and⟨⋅,⋅⟩{\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle } a semi-definite sesquilinear form, then the function:‖x‖=⟨x,x⟩{\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}}makes sense and satisfies all the properties of norm except that‖x‖=0{\displaystyle \|x\|=0} does not implyx=0{\displaystyle x=0} (such a functional is then called asemi-norm). We can produce an inner product space by considering the quotientW=V/{x:‖x‖=0}.{\displaystyle W=V/\{x:\|x\|=0\}.} The sesquilinear form⟨⋅,⋅⟩{\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle } factors throughW.{\displaystyle W.}

This construction is used in numerous contexts. TheGelfand–Naimark–Segal construction is a particularly important example of the use of this technique. Another example is the representation ofsemi-definite kernels on arbitrary sets.

Nondegenerate conjugate symmetric forms

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Main article:Pseudo-Euclidean space

Alternatively, one may require that the pairing be anondegenerate form, meaning that for all non-zerox≠0{\displaystyle x\neq 0} there exists somey{\displaystyle y} such that⟨x,y⟩≠0,{\displaystyle \langle x,y\rangle \neq 0,} thoughy{\displaystyle y} need not equalx{\displaystyle x}; in other words, the induced map to the dual spaceV→V∗{\displaystyle V\to V^{*}} is injective. This generalization is important indifferential geometry: a manifold whose tangent spaces have an inner product is aRiemannian manifold, while if this is related to nondegenerate conjugate symmetric form the manifold is apseudo-Riemannian manifold. BySylvester'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 withnonzero 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 inMinkowski 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 fourdimensions and indices 3 and 1 (assignment of"+" and "−" to themdiffers depending on conventions).

Purely algebraic statements (ones that do not use positivity) usually only rely on the nondegeneracy (the injective homomorphismV→V∗{\displaystyle V\to V^{*}}) and thus hold more generally.

Related products

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The term "inner product" is opposed toouter product (tensor product), which is a slightly more general opposite. Simply, in coordinates, the inner product is the product of a1×n{\displaystyle 1\times n}covector with ann×1{\displaystyle n\times 1} vector, yielding a1×1{\displaystyle 1\times 1} matrix (a scalar), while the outer product is the product of anm×1{\displaystyle m\times 1} vector with a1×n{\displaystyle 1\times n} covector, yielding anm×n{\displaystyle m\times n} matrix. 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 thetrace of the outer product (trace only being properly defined for square matrices). In an informal summary: "inner is horizontal times vertical and shrinks down, outer is vertical times horizontal and expands out".

More abstractly, the outer product is the bilinear mapW×V∗→hom⁡(V,W){\displaystyle W\times V^{*}\to \hom(V,W)} 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 mapV∗×V→F{\displaystyle V^{*}\times V\to F} 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 theinterior product andexterior product, which are instead operations onvector fields anddifferential forms, or more generally on theexterior algebra.

As a further complication, ingeometric algebra the inner product and theexterior (Grassmann) product are combined in the geometric product (the Clifford product in aClifford 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 theouter product (alternatively,wedge product). The inner product is more correctly called ascalar product in this context, as the nondegenerate quadratic form in question need not be positive definite (need not be an inner product).

See also

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Notes

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  1. ^Also called, rarely, aHausdorff pre-Hilbert space,[1][2] titled afterHausdorff spaces andHilbert spaces.
  2. ^Combining thelinearity in the first argument property with theconjugate symmetry property provesconjugate-linearity in the second argument:⟨x,by⟩=⟨x,y⟩b¯{\textstyle \langle x,by\rangle =\langle x,y\rangle {\overline {b}}}. This is how the inner product was originally defined and is used in most mathematical contexts. A different convention has been adopted in theoretical physics and quantum mechanics, originating in thebra-ket notation ofPaul Dirac, where the inner product is taken to belinear in the second argument andconjugate-linear in the first argument; this convention is used in many other domains such as engineering and computer science.
  3. ^⟨0,x⟩=⟨x−x,x⟩=⟨x,x⟩−⟨x,x⟩=0{\displaystyle \langle \mathbf {0} ,x\rangle =\langle x-x,x\rangle =\langle x,x\rangle -\langle x,x\rangle =0} where the right-hand side of the second equality comes from the first argument linearity.⟨x,0⟩=0{\displaystyle \langle x,\mathbf {0} \rangle =0} is also similarly proved by using the conjugate symmetry and the first argument linearity.
  4. ^⟨x,x⟩=⟨x,x⟩¯{\displaystyle \langle x,x\rangle ={\overline {\langle x,x\rangle }}} so it is a real number. Forx≠0{\displaystyle x\neq \mathbf {0} }, it is a positive real number by the positive-definiteness. Forx=0{\displaystyle x=\mathbf {0} }, it is zero by the 1st basic property above. So,⟨x,x⟩{\displaystyle \langle x,x\rangle } is real and nonnegative.
  5. ^By the 2nd basic property above and the positive-definiteness.

References

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  1. ^abcTrèves 2006, pp. 112–125.
  2. ^Schaefer & Wolff 1999, pp. 40–45.
  3. ^Moore, Gregory H. (1995)."The axiomatization of linear algebra: 1875-1940".Historia Mathematica.22 (3):262–303.doi:10.1006/hmat.1995.1025.
  4. ^Schaefer & Wolff 1999, pp. 36–72.
  5. ^Jain, P. K.; Ahmad, Khalil (1995)."5.1 Definitions and basic properties of inner product spaces and Hilbert spaces".Functional Analysis (2nd ed.). New Age International. p. 203.ISBN 81-224-0801-X.
  6. ^Prugovečki, Eduard (1981)."Definition 2.1".Quantum Mechanics in Hilbert Space (2nd ed.). Academic Press. pp. 18ff.ISBN 0-12-566060-X.
  7. ^Schaefer & Wolff 1999, p. 44.
  8. ^Ouwehand, Peter (November 2010)."Spaces of Random Variables"(PDF).AIMS. Archived fromthe original(PDF) on 2017-09-05. Retrieved2017-09-05.
  9. ^Siegrist, Kyle (1997)."Vector Spaces of Random Variables".Random: Probability, Mathematical Statistics, Stochastic Processes. Retrieved2017-09-05.
  10. ^Bigoni, Daniele (2015)."Appendix B: Probability theory and functional spaces"(PDF).Uncertainty Quantification with Applications to Engineering Problems (PhD). Technical University of Denmark. Retrieved2017-09-05.
  11. ^Apostol, Tom M. (1967)."Ptolemy's Inequality and the Chordal Metric".Mathematics Magazine.40 (5):233–235.doi:10.2307/2688275.JSTOR 2688275.
  12. ^abRudin 1991, pp. 306–312.
  13. ^Rudin 1991

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