Dual Space

There are two types of dual spaces: the algebraic dual space, and the continuous dual space. The algebraic dual space is defined for all vector spaces. When defined for a Topological Vector Space there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a ''continuous dual space''. ALGEBRAIC DUAL SPACE to be the set of all Linear Functional s on ''V'', i.e., Scalar -valued Linear Map s on ''V'' (in this context, a "scalar" is a member of the base-field ''F''). ''V''--- itself becomes a vector space over ''F'' under the following definition of addition and scalar multiplication: : $(\phi + \psi )( x ) = \phi ( x ) + \psi ( x ) \,$ : $( a \phi ) ( x ) = a \phi ( x ) \,$ , ''a'' in ''F'' and ''x'' in ''V''. , Covariant vectors, covectors or ''' One-form s'''. The finite dimensional case has the same dimension as ''V''; is given by : $\mathbf{e}^i (\mathbf{e}_j)= \left\{\begin{matrix} 1, & \mbox{if }i = j \ 0, & \mbox{if } i e j \end{matrix} ight.$ In the case of R², its basis is ''B''={'''e'''₁=(1,0),'''e'''₂=(0,1)}. Then, '''e'''¹, and '''e'''² are one-forms (functions which map a vector to a scalar) such that '''e'''¹('''e'''₁)=1, '''e'''¹('''e'''₂)=0, '''e'''²('''e'''₁)=0, and '''e'''²('''e'''₂)=1. (Note: The superscript here is an index, not an exponent.) Concretely, if we interpret R^''n'' as the space of columns of ''n'' Real Number s, its dual space is typically written as the space of ''rows'' of ''n'' real numbers. Such a row acts on R^''n'' as a linear functional by ordinary Matrix Multiplication . can be intuitively represented as collections of parallel lines. Such a collection of lines can be applied to a vector to yield a number in the following way: one counts how many of the lines the vector crosses. The infinite dimensional case If ''V'' is not finite-dimensional but has a basisSeveral assertions in this article require the elements e^''α'' (''α''∈''A'') of the dual space, but they will not form a basis. Consider, for instance, the space R^∞, whose elements are those of R^∞ is countably infinite, whereas R^'''N''' does not have a countable basis. This observation generalizes to any infinite dimensional vector space ''V'' over any field F: a choice of basis {'''e'''_''α'':''α''∈''A''} identifies ''V'' with the space (F^''A'')₀ of functions ''f'':''A''→F such that ''f''_''α''=''f''(''α'') is nonzero for only finitely many ''α''∈''A'', where such a function ''f'' is identified with the vector : $\sum_{\alpha\in A} f_\alpha\mathbf{e}_\alpha$ in ''V'' (the sum is finite by the assumption on ''f'' and any ''v''∈''V'' may be written in this way by the definition of a basis). The dual space of ''V'' may then be identified with the space F^''A'' of ''all'' functions from ''A'' to F: a linear functional ''T'' on ''V'' is uniquely determined by the values ''θ''_''α''=''T''('''e'''_''α'') it takes on the basis of ''V'', and any function ''θ'':''A''→F (with ''θ''(''α'')=''θ''_''α'') defines linear functional ''T'' on ''V'' by : $T\biggl(\sum_{\alpha\in A} f_\alpha e_\alpha\biggr) = \sum_{\alpha\in A} heta_\alpha f_\alpha.$ Again the sum is finite because ''f''_''α'' is nonzero for only finitely many ''α''. Note that (F^''A'')₀ may be identified (essentially by definition) with the Direct Sum of infinitely many copies of F (viewed as a 1-dimensional vector space over itself) indexed by ''A'', i.e., there are linear isomorphisms : $V\cong (\mathbb F^A)_0\cong\bigoplus_{\alpha\in A} \mathbb{F}.$ On the other hand F^''A'' is (again by definition), the Direct Product of infinitely many copies of F indexed by ''A'', and so the identification \cong \biggl(\bigoplus_{\alpha\in A}\mathbb{F}\biggr)^---\cong \prod_{\alpha\in A}\mathbb{F}^---\cong\prod_{\alpha\in A}\mathbb{F}\cong \mathbb{F}^A is a special case of a General Result relating direct sums (of modules) to direct products. Thus if the basis is infinite, then there are ''always'' more vectors in the dual space than the original vector space. This is in marked contrast to the case of the continuous dual space, discussed below, which may be isomorphic to the original vector space even if the latter is infinite-dimensional. Bilinear products and dual spaces , but the isomorphism is not Natural and depends on the basis of ''V'' we started out with. In fact, any isomorphism Φ from ''V'' to ''V''--- defines a unique non-degenerate Bilinear Form on ''V'' by : $\langle v,w angle = (\Phi (v))(w) \,$ . Injection into the double-dual ---, defined by $(\Psi(v))(\phi) = \phi(v)$ for all ''v'' in ''V'', $\phi$ in ''V''---. This map $\Psi$ is always Injective ; it is an isomorphism if and only if ''V'' is finite-dimensional. (Infinite-dimensional Hilbert spaces are not a counterexample to this, as they are isomorphic to their continuous duals, not to their algebraic duals.) Transpose of a linear map : ''W''--- $o$ ''V''--- by (arphi) = arphi \circ f \, . In that case, $f^--- (\phi)$ is also known as the '' Pullback '' of $\phi$ by ''f''. produces an Injective linear map between the space of linear operators from ''V'' to ''W'' and the space of linear operators from ''W''--- to ''V''---; this homomorphism is an Isomorphism if and only if ''W'' is finite-dimensional. If ''V'' = ''W'' then the space of linear maps is actually an Algebra under Composition Of Maps , and the assignment is then an Antihomomorphism of algebras, meaning that (''fg'')--- = ''g''---''f''---. In the language of Category Theory , taking the dual of vector spaces and the transpose of linear maps is therefore a Contravariant Functor from the category of vector spaces over ''F'' to itself. Note that one can identify (''f''---)--- with ''f'' using the natural injection into the double dual. is represented by the Transpose matrix ^t''A'' with respect to the dual bases of ''W''--- and ''V''---, hence the name. Alternatively, as ''f'' is represented by ''A'' acting on the left on column vectors, ''f''--- is represented by the same matrix acting by the right on row vectors. These points of view are related by the canonical inner product on R^''n'', which identifies the space of column vectors with the dual space of row vectors. CONTINUOUS DUAL SPACE ''See main article Continuous Dual Space '' , denoted ''V'' ′. For any ''finite-dimensional'' normed vector space or topological vector space, such as Euclidean ''n-''space , the continuous dual and the algebraic dual coincide. This is however false for any infinite-dimensional normed space. In topological contexts sometimes ''V''--- may also be used for just the continuous dual space and the continuous dual may just be called the ''dual''.
	:<math>\\phi \	\sup \{ \phi ( x ) : \x\ \le 1 \}</math>
	:<math>\\mathbf{a}\ P	\left ( \sum_{n=0}^\infty a_n^p ight) ^{1/p}</math>
	In Analogy With The Case Of The Algebraic Double Dual, There Is Always A Naturally Defined Injective Continuous Linear Operator &Psi&nbsp:&nbsp''V'' &rarr ''V'' &prime&prime From ''V'' Into Its Continuous Double Dual ''V'' &prime&prime This Map Is In Fact An	"http://wwwinformationdelightinfo/information/entry/isometry" class="copylinks">Isometry , meaning &Psi(''x'') = ''x'' for all ''x'' in ''V'' Spaces for which the map &Psi is a Bijection are called Reflexive

	CATEGORIES ABOUT DUAL SPACE

	linear algebra

	functional analysis

	duality theories

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Dual Space