Dual pairing

In mathematics , the dual pairing is a mapping that assigns a number to a vector and a linear functional . It represents a generalization of the scalar product .

The goal is to use mathematical terms that come from a scalar product (such as the question of whether two vectors are perpendicular to each other) in spaces in which one cannot define a scalar product (and therefore also cannot measure angles ). The disadvantage that arises is that the two vectors, the scalar product of which is calculated (in order to obtain their angle, for example), come from different vector spaces.

In physics , approaches to dual pairing appear in the Bra-Ket formalism, for example .

definition

Let there be a - vector space and the corresponding dual space . The image ${\ displaystyle X}$${\ displaystyle K}$${\ displaystyle X ^ {\ prime}}$

${\ displaystyle \ langle \ cdot, \ cdot \ rangle: X \ times X ^ {\ prime} \ rightarrow K, \ quad (x, l) \ mapsto \ langle x, l \ rangle: = l (x)}$

is called dual pairing.

If the vector space under consideration has a topological structure, one usually means the topological dual space, i.e. the space of continuous linear functionals. ${\ displaystyle X ^ {\ prime}}$

properties

The dual pairing on standardized spaces

If there is a normalized space , then applies ${\ displaystyle X}$

${\ displaystyle (\ forall x \ in X: \ langle x, l \ rangle = 0) \ Rightarrow l = 0}$
${\ displaystyle (\ forall l \ in X ^ {\ prime}: \ langle x, l \ rangle = 0) \ Rightarrow x = 0}$,

where the second statement is a corollary from Hahn-Banach's Theorem for normalized spaces. In this case the dual pairing is a non-degenerate bilinear map .

In normalized spaces there is an inequality that represents a generalization of the Cauchy-Schwarz inequality . Are and the operator norm of , then is ${\ displaystyle x \ in X, l \ in X ^ {\ prime}}$${\ displaystyle \ | l \ | _ {Op}}$${\ displaystyle l}$

${\ displaystyle {\ Big |} {\ Big \ langle} {\ frac {x} {\ | x \ |}}, l {\ Big \ rangle} {\ Big |} = {\ Big |} l {\ Big (} {\ frac {x} {\ | x \ |}} {\ Big)} {\ Big |} \ leq \ | l \ | _ {Op}}$

and therefore

${\ displaystyle | \ langle x, l \ rangle | \ leq \ | x \ | \ cdot \ | l \ | _ {Op}.}$

The dual pairing on Hilbert spaces

If a Hilbert space is , it is because of the Fréchet-Riesz representation theorem . If, in addition, is a real vector space, then the dual pairing in this case is identical to the scalar product of the Hilbert space. For complex Hilbert spaces it should be noted that the dual pairing is bilinear, in contrast to the scalar product, which is only sesquilinear . ${\ displaystyle X}$ ${\ displaystyle X \ cong X ^ {\ prime}}$${\ displaystyle X}$

In order to avoid confusion with a (possibly sesquilinear) inner product, the notation for the dual pairing is sometimes reserved in the literature and used for the inner product . You then get the relationship ${\ displaystyle \ langle \ cdot, \ cdot \ rangle}$${\ displaystyle (\ cdot, \ cdot)}$

${\ displaystyle \ langle x, y \ rangle = (x, {\ overline {y}}).}$

The notation of the dual pairing is compatible with certain calculation rules that are known for adjoint operators on Hilbert spaces. If a Hilbert space, a linear operator and the adjoint operator, then is ${\ displaystyle X}$${\ displaystyle T \ colon X \ rightarrow X}$${\ displaystyle T ^ {*} \ colon X \ rightarrow X}$

${\ displaystyle (Tx, y) = (x, T ^ {*} y)}$

for all in the domain of and all in the domain of . If there is no longer a Hilbert space, then one obtains (since in this case no analogue to the Riesz isomorphism exists) as adjoint to an operator on the dual spaces and it applies ${\ displaystyle x}$${\ displaystyle T}$${\ displaystyle y}$${\ displaystyle T ^ {*}}$${\ displaystyle X}$${\ displaystyle T}$${\ displaystyle T ^ {\ prime} \ colon X ^ {\ prime} \ rightarrow X ^ {\ prime}}$

${\ displaystyle \ langle Tx, y \ rangle = y (Tx) = (T ^ {\ prime} y) (x) = \ langle x, T ^ {\ prime} y \ rangle}$

for all in the domain of and all in the domain of . ${\ displaystyle x}$${\ displaystyle T}$${\ displaystyle y}$${\ displaystyle T ^ {\ prime}}$

Gelfand triples

definition

A coexistence of dual pairing and scalar products is obtained, for example, in the following situation. Consider a Hilbert space , and a part space , which is provided with a topology that is finer than the induced subspace topology is such that the inclusion mapping is continuous. ${\ displaystyle H}$ ${\ displaystyle N}$${\ displaystyle N}$${\ displaystyle i \ colon N \ rightarrow H}$

Again, due to the Riesz isomorphism, one can identify with its topological dual space. There is also a dual map for inclusion mapping ${\ displaystyle H}$

${\ displaystyle i ^ {\ prime}: H ^ {\ prime} \ rightarrow N ^ {\ prime}.}$

Often one demands that a dense subspace of is, since then the mapping is injective and becomes an embedding . One therefore writes the chain of inclusion ${\ displaystyle N}$${\ displaystyle H}$${\ displaystyle i ^ {\ prime}}$

${\ displaystyle N \ subset H = H ^ {\ prime} \ subset N ^ {\ prime}}$,

what is called the Gelfand triple, named after IM Gelfand . Here, too, one can consider a dual pairing for , but only then with the scalar product in the relationship ${\ displaystyle N, N ^ {\ prime}}$${\ displaystyle H}$

${\ displaystyle \ langle x, y \ rangle = (x, {\ overline {y}})}$

can stand if an element is from . ${\ displaystyle y}$${\ displaystyle H ^ {\ prime}}$

example

An important Gelfand triple from white noise analysis is the triple

${\ displaystyle {\ mathcal {S}} (\ mathbb {R}) \ subset L ^ {2} (\ mathbb {R}) \ subset {\ mathcal {S}} (\ mathbb {R}) ^ {\ prime},}$

where is the space of rapidly falling functions ( Schwartz space ) and its topological dual space, the space of tempered distributions . is the Hilbert space of the quadratically integrable functions with respect to the Lebesgue measure. The Schwartz space is a dense subspace and it is a complete metric space, but no scalar product can be defined on it that creates its topology. ${\ displaystyle {\ mathcal {S}} (\ mathbb {R})}$${\ displaystyle {\ mathcal {S}} (\ mathbb {R}) ^ {\ prime}}$${\ displaystyle L ^ {2} (\ mathbb {R})}$

You can now each element in a temperature-controlled distribution regarded by the picture ${\ displaystyle f}$${\ displaystyle L ^ {2}}$

${\ displaystyle {\ overline {f}} \ colon {\ mathcal {S}} \ rightarrow \ mathbb {R}, \ quad g \ mapsto \ int _ {\ mathbb {R}} g (x) f (x) dx}$

defined (the finiteness of the integral is a consequence of the Cauchy-Schwarz inequality ). One quickly sees that the scalar product and dual pairing for all elements agree with this definition , if it can be induced by a function that can be integrally integrated with the square . This is not possible for other tempered distributions (so-called singular distributions), for example for the delta distribution , since the expression ${\ displaystyle {\ overline {f}} \ in {\ mathcal {S}} (\ mathbb {R}) ^ {\ prime}, g \ in {\ mathcal {S}} (\ mathbb {R})}$${\ displaystyle {\ overline {f}}}$${\ displaystyle f}$

${\ displaystyle \ int _ {\ mathbb {R}} g (x) \ delta (x) dx}$

is purely formal and does not represent a Lebesgue integral.

The annihilator room

With the help of the dual pairing, a generalization of the orthogonal complement of a set can be defined for any vector spaces , the so-called annihilator space${\ displaystyle X}$${\ displaystyle S \ subset X}$

${\ displaystyle S ^ {0} = \ lbrace f \ in X ^ {\ prime} \ mid f (S) = \ lbrace 0 \ rbrace \ rbrace.}$

The dual pairing in physics

In physics, the dual pairing is usually defined differently, so that the order of vector space and dual space is reversed. You get

${\ displaystyle \ langle \ cdot, \ cdot \ rangle: X ^ {\ prime} \ times X \ rightarrow K, (l, x) \ mapsto \ langle l, x \ rangle = l (x).}$

One reason that speaks for this definition may be the similarity to the Euclidean scalar product. There you can see vectors as column vectors and the associated functionals as row vectors. Then the matrix multiplication applies with the calculation rules

${\ displaystyle \ langle \ cdot, \ cdot \ rangle: (\ mathbb {R} ^ {n}) ^ {\ prime} \ times \ mathbb {R} ^ {n} \ rightarrow \ mathbb {R}, (x , y) \ mapsto \ langle x, y \ rangle = (x_ {1}, \ ldots, x_ {n}) {\ begin {pmatrix} y_ {1} \\\ vdots \\ y_ {n} \ end { pmatrix}}.}$

Bra-ket notation

In the Bra-Ket formalism, which is often used in quantum mechanics , vectors are written as Ket vectors in the form and elements of the dual space as Bra vectors in the form . If one compares this notation with the above remark about the Euclidean scalar product, one recognizes that the same idea is based here, namely that one can formally write a scalar product as the product of a functional and a vector. ${\ displaystyle | y \ rangle}$${\ displaystyle \ langle x |}$

However, it should be noted that this is not based on a dual pairing, since the vector spaces in quantum mechanics are often complex spaces and the scalar product is therefore sesquilinear. Nevertheless, this notation, which is related to the dual pairing, is useful because it enables intuitive computing with vectors, functionals and scalar products.

literature

• Nobuaki Obata: White Noise Calculus and Fock Space ("Lecture notes in mathematics; 1577"). Springer Verlag, Berlin 1994, ISBN 3-540-57985-0 .