In mathematics, the complex conjugate of a complex vector space V {\displaystyle V\,} is a complex vector space V ¯ {\displaystyle {\overline {V}}} that...

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in...

mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an...

In mathematics, a complex structure on a real vector space V {\displaystyle V} is an automorphism of V {\displaystyle V} that squares to the minus identity...

the conjugate transpose, also known as the Hermitian transpose, of an m × n {\displaystyle m\times n} complex matrix A {\displaystyle \mathbf {A} } is...

In mathematics, a function f : V → W {\displaystyle f:V\to W} between two complex vector spaces is said to be antilinear or conjugate-linear if f ( x...

called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional...

mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces. The prototype...

complexification of a vector space V over the field of real numbers (a "real vector space") yields a vector space VC over the complex number field, obtained...

\mathbb {C} ^{2}} be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements 1 and j. A vector in C 2 {\displaystyle...

product of a complex vector a ⋅ a = a H a {\displaystyle \mathbf {a} \cdot \mathbf {a} =\mathbf {a} ^{\mathsf {H}}\mathbf {a} } , involving the conjugate transpose...

operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector spaces and complex vector spaces are...

In mathematics, a symplectic vector space is a vector space V {\displaystyle V} over a field F {\displaystyle F} (for example the real numbers R {\displaystyle...

is a group and Π is a representation of it over the complex vector space V, then the complex conjugate representation Π is defined over the complex conjugate...

equals the inverse. Over a complex vector space, one often works with sesquilinear forms (conjugate-linear in one argument) instead of bilinear forms. The...

mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle...

the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: A  is Hermitian ⟺ a i j = a j i ¯ {\displaystyle A{\text{...

In mathematics, a Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the metric induced by the...

{\displaystyle {\overline {\mathfrak {g}}}} is a complex Lie algebra with the same underlying real vector space but with i = − 1 {\displaystyle i={\sqrt {-1}}}...

number and zero. In vector spaces, the Euclidean norm is a measure of magnitude used to define a distance between two points in space. In physics, magnitude...

{p} _{n}\}} is a set of n {\displaystyle n} mutually conjugate vectors with respect to A {\displaystyle \mathbf {A} } , i.e. p i T A p j = 0 {\displaystyle...

and a vector of the original vector space. A linear map A from a vector space of dimension n into a vector space of dimension m maps a column vector x =...

complex conjugate vector space to V . {\displaystyle V.} By the universal property of tensor products these are in one-to-one correspondence with complex linear...

In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance...

(possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear...

In mathematics, a complex representation is a representation of a group (or that of Lie algebra) on a complex vector space. Sometimes (for example in physics)...

1) (Rudin 1987, Def 17.7). The Hardy spaces can also be viewed as closed vector subspaces of the complex Lp spaces on the unit circle T = { z ∈ C : | z...

analysis, a Banach space (/ˈbɑː.nʌx/, Polish pronunciation: [ˈba.nax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric...

algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping...

elements of a complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight...