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Concept
Complex conjugate of a vector space
Summary
In mathematics, the complex conjugate of a complex vector space is a complex vector space , which has the same elements and additive group structure as but whose scalar multiplication involves conjugation of the scalars. In other words, the scalar multiplication of satisfies
where is the scalar multiplication of and is the scalar multiplication of
The letter stands for a vector in is a complex number, and denotes the complex conjugate of
More concretely, the complex conjugate vector space is the same underlying vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure (different multiplication by ).
If and are complex vector spaces, a function is antilinear if
With the use of the conjugate vector space , an antilinear map can be regarded as an ordinary linear map of type The linearity is checked by noting:
Conversely, any linear map defined on gives rise to an antilinear map on
This is the same underlying principle as in defining opposite ring so that a right -module can be regarded as a left -module, or that of an so that a contravariant functor can be regarded as an ordinary functor of type
A linear map gives rise to a corresponding linear map which has the same action as Note that preserves scalar multiplication because
Thus, complex conjugation and define a functor from the of complex vector spaces to itself.
If and are finite-dimensional and the map is described by the complex matrix with respect to the bases of and of then the map is described by the complex conjugate of with respect to the bases of and of
The vector spaces and have the same dimension over the complex numbers and are therefore isomorphic as complex vector spaces. However, there is no natural isomorphism from to
The double conjugate is identical to
Given a Hilbert space (either finite or infinite dimensional), its complex conjugate is the same vector space as its continuous dual space
There is one-to-one antilinear correspondence between continuous linear functionals and vectors.
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Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.
Riesz representation theorem
The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism.