Dual space

In the mathematical subfield of linear algebra , the (algebraic) dual space of a vector space over a field is the vector space of all linear mappings from to . These linear mappings are sometimes called co- vectors . ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle K}$

If the vector space is finite dimensional, it has the same dimension as its dual space. The two vector spaces are therefore isomorphic . ${\displaystyle V}$

In functional analysis , one considers the topological dual space of a (generally infinite-dimensional) topological vector space . This consists of all continuous linear functionals . The dual space of a dual space is called bidual space .

The algebraic dual space

Definition and concept formation

A vector space  over a body denotes the associated dual space, that is, the set of all linear mappings from to  . Depending on the context, its elements are also called functional , linear forms or 1-forms . The language of tensor algebra is particularly popular in physics ; then the elements of  contravariant are called , those of  covariant vectors or also covectors . The mapping is a non-degenerate bilinear form and is called dual pairing . ${\displaystyle V}$  ${\displaystyle K}$ ${\displaystyle V^{*}}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle V^{*}}$ ${\displaystyle V\times V^{*}\to K,\ (x,f)\mapsto \langle x,f\rangle :=f(x)}$

Dual space as vector space

Due to the following definition of addition and scalar multiplication of  on there  is itself a vector space over the body   . ${\displaystyle K}$ ${\displaystyle V^{*}}$ ${\displaystyle V^{*}}$ ${\displaystyle K}$

For this purpose the vectorial addition

${\displaystyle +\colon V^{*}\times V^{*}\rightarrow V^{*}}$through for everyone ${\displaystyle \left(f+g\right)(x):=f(x)+g(x)}$ ${\displaystyle x\in V,f,g\in V^{*}}$

and the scalar multiplication

${\displaystyle \cdot \colon K\times V^{*}\rightarrow V^{*}}$through for everyone ${\displaystyle (\alpha f)\left(x\right):=\alpha f(x)}$ ${\displaystyle x\in V,f\in V^{*},\,\alpha \in K}$

Are defined.

Basis of the dual space

If a -dimensional vector space is -dimensional. So it applies . ${\displaystyle V}$ ${\displaystyle n}$ ${\displaystyle V^{*}}$ ${\displaystyle n}$ ${\displaystyle \dim _{K}V^{*}=\dim _{K}V=n}$

Be a basis of , then is called with ${\displaystyle X=\left\{x_{i}\right\}_{i=1,2,\dotsc ,n}}$ ${\displaystyle V}$ ${\displaystyle X^{*}=\left\{x_{i}^{*}\right\}_{i=1,2,\dotsc ,n}}$

${\displaystyle x_{i}^{*}:}$	${\displaystyle V\,\rightarrow \,K}$	${\displaystyle \,}$	${\displaystyle \,}$ linear and
	${\displaystyle x_{i}^{*}(x_{j})\,}$	${\displaystyle =}$	${\displaystyle {\begin{cases}1,&{\text{falls}}\;i=j\\0,&{\text{falls}}\;j\neq i\end{cases}}}$

the dual basis to the basis and is a basis of the dual space . With the help of dual pairing, the effect of dual basis vectors on basis vectors can be clearly written using the Kronecker delta : ${\displaystyle X}$ ${\displaystyle V^{*}}$ ${\displaystyle x_{i}^{*}\in V^{*}}$ ${\displaystyle x_{j}\in V}$

${\displaystyle \langle x_{j},x_{i}^{*}\rangle =\delta _{ij}}$.

By identifying each linear form of the algebraic dual space with its core , i.e. the solution set of the homogeneous linear equation , one arrives at a duality between points and hyperplanes of the projective space in projective geometry . This duality is shown in the article " Projective Coordinate System ". ${\displaystyle f}$ ${\displaystyle f(x)=0}$

If, on the other hand, an infinite-dimensional vector space, then in general no dual basis can be constructed in this way. Namely, be a basis of the infinite-dimensional vector space . Then one can look at the linear mapping . This is an element of the dual space , but it cannot be represented as a finite linear combination of   . Therefore they do not form a generating system of . ${\displaystyle V}$ ${\displaystyle (x_{i})_{i\in I}}$ ${\displaystyle V}$ ${\displaystyle f\colon V\to K,f(x_{i})=1\,\forall i\in I}$ ${\displaystyle V^{*}}$ ${\displaystyle x_{i}^{*}}$ ${\displaystyle x_{i}^{*}}$ ${\displaystyle V^{*}}$

Dual mapping

If there is a linear mapping between -Vector spaces and , then is through ${\displaystyle F\colon V\to W}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle W}$

${\displaystyle F^{\ast }\colon W^{\ast }\to V^{\ast },\quad f\mapsto F^{\ast }(f)=f\circ F}$

a linear mapping between the dual spaces and given. It is called the too dual mapping . ${\displaystyle W^{\ast }}$ ${\displaystyle V^{\ast }}$ ${\displaystyle F}$

If -linear mappings, then ${\displaystyle F,G\colon V\to W}$ ${\displaystyle K}$

${\displaystyle (F+G)^{\ast }=F^{\ast }+G^{\ast }}$

as well as for everyone ${\displaystyle \alpha \in K}$

${\displaystyle (\alpha F)^{\ast }=\alpha \cdot F^{\ast }}$.

A linear mapping is thus given by the assignment . ${\displaystyle F\mapsto F^{\ast }}$ ${\displaystyle K}$ ${\displaystyle \operatorname {Hom} (V,W)\to \operatorname {Hom} (W^{\ast },V^{\ast })}$

If is an injective linear map, then the dual map is surjective . If, on the other hand, is surjective, then is injective. ${\displaystyle F}$ ${\displaystyle F^{\ast }}$ ${\displaystyle F}$ ${\displaystyle F^{\ast }}$

Is another -Vector space and are and linear, then holds ${\displaystyle U}$ ${\displaystyle K}$ ${\displaystyle F\colon U\to V}$ ${\displaystyle G\colon V\to W}$

${\displaystyle (G\circ F)^{\ast }=F^{\ast }\circ G^{\ast }}$.

Double room

The dual space of the dual space of a vector space is called the bidual space and is denoted by. The elements of are therefore linear mappings that assign scalars from to the functional . For each , the mapping that assigns the scalar to each is such a mapping, that is, it is true . ${\displaystyle (V^{\ast })^{\ast }}$ ${\displaystyle V^{\ast }}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle V^{\ast \ast }}$ ${\displaystyle V^{\ast \ast }}$ ${\displaystyle f\in V^{\ast }}$ ${\displaystyle K}$ ${\displaystyle v\in V}$ ${\displaystyle \Phi _{v}}$ ${\displaystyle f\in V^{\ast }}$ ${\displaystyle f(v)}$ ${\displaystyle \Phi _{v}\in V^{\ast \ast }}$

The image

${\displaystyle \Phi \colon V\to V^{\ast \ast },v\mapsto \Phi _{v}}$ With ${\displaystyle \Phi _{v}(f)=f(v)}$

is linear and injective. Therefore it can always be identified with a subspace of . One calls the natural or canonical embedding of space in its bidual space. ${\displaystyle V}$ ${\displaystyle V^{\ast \ast }}$ ${\displaystyle \Phi }$

If is finite-dimensional, then applies . In this case is even bijective and is called canonical isomorphism between and . ${\displaystyle V}$ ${\displaystyle \dim _{K}V=\dim _{K}V^{\ast }=\dim _{K}V^{\ast \ast }}$ ${\displaystyle \Phi }$ ${\displaystyle V}$ ${\displaystyle V^{\ast \ast }}$

The topological dual space

If the underlying vector space is  a topological vector space , one can also consider the topological dual space in addition to the algebraic space. This is the set of all continuous linear functionals and is usually  denoted by. The distinction between algebraic and topological dual space is only important if there is an infinite-dimensional space, since all linear operators that are defined on a finite-dimensional topological vector space are also continuous. Thus the algebraic and the topological dual space are identical. When a dual space is mentioned in connection with topological vector spaces, the topological dual space is usually meant. The study of these dual spaces is one of the main areas of functional analysis . ${\displaystyle V}$ ${\displaystyle V\,'}$ ${\displaystyle V}$

Topological dual space of a standardized space

The spaces considered in functional analysis often have a topology that is induced by a norm. In this case the topological dual space is also a normalized vector space with the operator norm . ${\displaystyle \|f\|=\sup _{\|x\|\leq 1}|f(x)|}$

Since the underlying field of a normalized space is either the field of real or complex numbers and is therefore complete, the dual space is also complete , i.e. a Banach space , regardless of whether it is complete. ${\displaystyle V\,'=L(V,K)}$ ${\displaystyle V}$

The (topological) dual space is particularly simple if it is a Hilbert space . According to a theorem, which M. Fréchet proved in 1907 for separable and F. Riesz in 1934 for general Hilbert spaces, a real Hilbert space and its dual space are isometrically isomorphic , see Fréchet-Riesz theorem . The interchangeability of space and dual space is particularly clear in Dirac's Bra-Ket spelling. This is especially used in quantum mechanics , because the quantum mechanical states are modeled by vectors in a Hilbert space. ${\displaystyle V}$

Since every finite-dimensional vector space over the real or complex numbers is isomorphic to a Hilbert space, finite-dimensional spaces are always dual to themselves.

The strong dual space of a locally convex space

If a locally convex space , as in the case of normalized spaces , denotes the space of continuous linear functionals. Marking a suitable topology on the dual space is more complex. The following definition is laid out in such a way that in the special case of the standardized space, the standard topology described above results on the dual space: ${\displaystyle E}$ ${\displaystyle E'}$

Is limited , defined 1 seminorm on . The amount of seminorms , wherein the limited amounts of passes, defines the so-called strong topology on . The strong topology is called the strong dual space and is sometimes called more precisely , where the subscript b stands for bounded (English: bounded , French: borné ). ${\displaystyle B\subset E}$ ${\displaystyle p_{B}(f):=\sup\{|f(x)|;x\in B\}}$ ${\displaystyle E'}$ ${\displaystyle p_{B}}$ ${\displaystyle B}$ ${\displaystyle E}$ ${\displaystyle E'}$ ${\displaystyle E'}$ ${\displaystyle E_{\text{b}}'}$

The weak - * - topology is also a frequently observed topology , but in the case of infinitely dimensional normed spaces this does not coincide with the above-described standard topology on the dual space. In the theory of locally convex spaces, the dual space usually means the strong dual space. ${\displaystyle E'}$

Double room

Since the dual space of  a normalized space is a Banach space according to the above , one can consider the dual space of the dual space, the so-called bidual space  . It is interesting here that there is a canonical embedding of in which is given by . (That means: every element of the original space  is naturally also an element of the dual space). If every element of the dual space can be represented by an element , more precisely if the canonical embedding is an isomorphism , then the Banach space is called reflexive . Reflexive spaces are easier to handle than non-reflexive ones, they are in some ways most similar to Hilbert spaces . In the non-reflexive case, the canonical embedding is no longer surjective but is still isometric , and one usually writes . Accordingly, every normalized space is contained in a Banach space; the transition from to the topological closure in is a possibility to form the completion of a standardized space. ${\displaystyle V'}$ ${\displaystyle V''}$ ${\displaystyle V}$ ${\displaystyle V''}$ ${\displaystyle v\mapsto \left(\left(f\colon V\rightarrow K\right)\mapsto f\left(v\right)\right)}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle V\to V''}$ ${\displaystyle V\subset V''}$ ${\displaystyle V}$ ${\displaystyle V''}$

An example of a non-reflexive space is the sequence space of all zero sequences with the maximum norm . The dual space can naturally be identified with the sequence space of limited sequences with the supremum norm. There are non-reflexive Banach spaces in which the canonical embedding is not an isomorphism, but there is another isomorphism between space and dual space. One example of this is the so-called James Room , after Robert C. James . ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{\infty }}$

Examples

In the following table, for a Banach space in the first column, a further Banach space is given in the second column, which is isometrically isomorphic to the dual space of in the sense of the duality given in the third column . More precisely this means: Each element is defined by the formula of duality and has a continuous linear functional . This gives you an image that is linear, bijective and isometric. ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle V}$ ${\displaystyle W\to V'}$

Banach space	Dual space	Dual pairing	comment
${\displaystyle c_{0}}$ = Space of zero sequences with the supremum norm	${\displaystyle \ell ^{1}}$ = Space of the absolutely summable sequences with the norm ${\displaystyle \|\cdot \|_{1}}$	${\displaystyle \langle (a_{n})_{n},(b_{n})_{n}\rangle =\sum _{n}a_{n}b_{n}}$	see episode space
${\displaystyle c}$ = Space of convergent sequences with the supremum norm	${\displaystyle \ell ^{1}}$ = Space of the absolutely summable sequences with the norm ${\displaystyle \|\cdot \|_{1}}$	${\displaystyle \langle (a_{n})_{n},(b_{n})_{n}\rangle =\sum _{n}a_{n}b_{n+1}+b_{1}\lim _{n}a_{n}}$
${\displaystyle \ell ^{1}}$ = Space of the absolutely summable sequences with the norm ${\displaystyle \|\cdot \|_{1}}$	${\displaystyle \ell ^{\infty }}$ = Space of limited consequences with the supremum norm ${\displaystyle \|\cdot \|_{\infty }}$	${\displaystyle \langle (a_{n})_{n},(b_{n})_{n}\rangle =\sum _{n}a_{n}b_{n}}$
${\displaystyle \ell ^{p}}$ = Space of the sequences with the norm that can be absolutely summed in the p th power ${\displaystyle \|\cdot \|_{p}}$	${\displaystyle \ell ^{q}}$ = Space of the sequences with the norm that can be absolutely summed in the q-th power ${\displaystyle \|\cdot \|_{q}}$	${\displaystyle \langle (a_{n})_{n},(b_{n})_{n}\rangle =\sum _{n}a_{n}b_{n}}$	${\displaystyle 1<p,q<\infty ,\,\,{\frac {1}{p}}+{\frac {1}{q}}=1}$
${\displaystyle K(H)}$= Space of compact operators on the Hilbert space ${\displaystyle H}$	${\displaystyle N(H)}$= Space of the nuclear operators on the Hilbert space ${\displaystyle H}$	${\displaystyle \langle A,B\rangle =Sp(AB)}$	see nuclear operator
${\displaystyle N(H)}$ = Space of the nuclear operators on the Hilbert space ${\displaystyle H}$	${\displaystyle B(H)}$= Space of bounded operators on the Hilbert space ${\displaystyle H}$	${\displaystyle \langle A,B\rangle =Sp(AB)}$	see nuclear operator
${\displaystyle N(E)}$ = Space of nuclear operators on ${\displaystyle E}$	${\displaystyle B(E,E'')}$ = Space of restricted operators ${\displaystyle E\to E''}$	${\displaystyle \langle \sum _{n}f_{n}(\cdot )x_{n},B\rangle =\sum _{n}(B(x_{n}))(f_{n})}$	${\displaystyle E}$Banach space with approximation property , see nuclear operator
${\displaystyle {\mathcal {S}}_{p}(H)}$= p- shadow class on the separable Hilbert space ${\displaystyle H}$	${\displaystyle {\mathcal {S}}_{q}(H)}$ = q-shadow class on the separable Hilbert space ${\displaystyle H}$	${\displaystyle \langle A,B\rangle =Sp(AB)}$	${\displaystyle 1<p,q<\infty ,\,\,{\frac {1}{p}}+{\frac {1}{q}}=1}$
${\displaystyle L^{p}(X,\mu )}$= Space of the functions integrable in the p-th power with the norm ${\displaystyle \|\cdot \|_{p}}$	${\displaystyle L^{q}(X,\mu )}$ = Space of the functions integrable in the q-th power with the norm ${\displaystyle \|\cdot \|_{q}}$	${\displaystyle \langle f,g\rangle =\int _{X}f(x)g(x)d\mu (x)}$	${\displaystyle (X,\mu )}$ Measure space , see duality of L p spaces ${\displaystyle 1<p,q<\infty ,\,\,{\frac {1}{p}}+{\frac {1}{q}}=1}$
${\displaystyle L^{1}(X,\mu )}$ = Space of integrable functions with the norm ${\displaystyle \|\cdot \|_{1}}$	${\displaystyle L^{\infty }(X,\mu )}$= Space of essentially limited , measurable functions with the norm ${\displaystyle \|\cdot \|_{\infty }}$	${\displaystyle \langle f,g\rangle =\int _{X}f(x)g(x)d\mu (x)}$	${\displaystyle (X,\mu )}$ ${\displaystyle \sigma }$- finite dimensional space
${\displaystyle C_{0}(X,{\mathbb {K} })}$= Space of continuous -valent functions that vanish in infinity, with the supremum norm ${\displaystyle {\mathbb {K} }}$	${\displaystyle M_{r}(X,{\mathbb {K} })}$= Space of regular signed / complex measures with total variation as the norm	${\displaystyle \langle f,\mu \rangle =\int _{X}f(x)d\mu (x)}$	${\displaystyle X}$ locally compact Hausdorff area

See also

• Dual operator

literature

• Siegfried Bosch : Linear Algebra. 3. Edition. Springer-Verlag, Berlin, Heidelberg 2006, ISBN 978-3-540-29884-7 .
• Dirk Werner : Functional Analysis. Springer-Verlag, 2005, ISBN 3-540-43586-7 .

Individual evidence

1. ^ Albrecht Beutelspacher : Lineare Algebra . 7th edition. Vieweg + Teubner Verlag , Wiesbaden 2010, ISBN 978-3-528-66508-1 , p. 140-141 .
2. ^ Helmut H. Schaefer : Topological Vector Spaces . Springer-Verlag, New York 1971, ISBN 0-387-05380-8 , pp. 22 .
3. ↑ Jürgen Elstrodt: Measure and integration theory . Springer-Verlag, Berlin 2009, p. 349 .