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ALGEBRAIC DUAL SPACE


  • to be the set of all Linear Functional s on V, i.e., Scalar -valued Linear Transformation 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 ) \,


Examples


If the Dimension of V is finite,
  • has the same dimension as V;

  • is given by


:
e^i (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={e1=(1,0),e2=(0,1)}.Then, e1 is a one-form (function which maps a vector to a scalar) such that e1(e1)=1, and e1(e2)=0. Similarity for e2.

Concretely, if we interpret R''n'' as 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.


  • and the dimension of V--- is greater than that of V. Consider for instance the space R(ω), whose elements are those Sequence s of real numbers which have only finitely many non-zero entries (dimension is countably infinite). The dual of this space is Rω, the space of all sequences of real numbers (dimension is uncountably infinite). Such a sequence (''a''''n'') is applied to an element (''x''''n'') of R(ω) to give the number ∑''n''''a''''n''''x''''n''.



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 (Ψ(v))(φ) = φ(v) for all v in V, φ in V---. This map Ψ is always Injective ; it is an isomorphism if and only if V is finite-dimensional.



Pullback of a linear map


  • : ''W''---→''V''--- by

  • (\phi ) = \phi \circ 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 Iff 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 pullback of linear maps is therefore a Contravariant Functor from the category of vector spaces over F to itself. Note also that (''f''---)--- = ''f''.


  • is represented by same matrix acting by multiplication on the right on row vectors. Using the canonical inner product on R''n'', one may identify the space with its dual, in which case the matrix can be represented by the Transposed Matrix t''A''.



CONTINUOUS DUAL SPACE


When dealing with Topological Vector Space s, one is typically only interested in the Continuous linear functionals from the space into the base field. This gives rise to the notion of the continuous dual space which is a linear subspace of the algebraic dual space. The continuous dual of a vector space ''V'' is denoted ''V''′. When the context is clear, 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 : ''V'' &rarr ''V''&nbsp<nowiki>''</nowiki> From ''V'' Into Its Continuous Double Dual ''V''&nbsp<nowiki>''</nowiki> This Map Is In Fact An "http://wwwinformationdelightinfo/encyclopedia/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