homogeneous. If the vector spaces are real then antilinearity is the same as linearity. Antilinear maps occur in quantum mechanics in the study of time...

and the automorphism is complex conjugation, a semilinear map is called an antilinear map. Similar notation (replacing Latin characters with Greek ones)...

bounded; the same is true of antilinear maps. The inverse of any antilinear (resp. linear) bijection is again an antilinear (resp. linear) bijection. The...

vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant map j : V → V {\displaystyle j\colon V\to V} which satisfies j 2 =...

{\displaystyle {\overline {V}}} , an antilinear map f : V → W {\displaystyle f:V\to W} can be regarded as an ordinary linear map of type V ¯ → W . {\displaystyle...

Algebra/Linear Transformations Additive map – Z-module homomorphism Antilinear map – Conjugate homogeneous additive map Bent function – Special type of Boolean...

spaces V {\textstyle V} over the complex numbers. In this context, any antilinear map φ : V → V {\textstyle \varphi :V\to V} that satisfies φ 2 = id V , {\displaystyle...

In mathematics, an antiunitary transformation is a bijective antilinear map U : H 1 → H 2 {\displaystyle U:H_{1}\to H_{2}\,} between two complex Hilbert...

linear map is unitary if it is surjective and isometric. (Use Polarization identity to show the only if part.) Antiunitary – Bijective antilinear map between...

algebras. These come in three types. There is an invariant complex antilinear map r: S → S with r2 = idS. The fixed point set of r is then a real vector...

coordinate then L : H → P {\displaystyle L:H\to P} is a linear map (resp. an antilinear map); either way, its kernel ker ⁡ L = L − 1 ( 0 ) = C ⊥ {\displaystyle...

complex α, homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity...

superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map * such that * respects the grading and [a,b]*=[b*,a*]. A unitary representation...

V.} A linear isometry (resp. an antilinear isometry) is an isometry that is also a linear map (resp. an antilinear map). For inner product spaces, the...

{\mathbb {C} }={\mathbb {R} }\oplus i{\mathbb {R} }\,} . The conjugation map is antilinear: σ ( λ z ) = λ ¯ σ ( z ) {\displaystyle \sigma (\lambda z)={\bar {\lambda...

particular, there exist additive maps that are not linear maps. Antilinear map – Conjugate homogeneous additive map Leslie Hogben (2013), Handbook of...

superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map from itself to itself which respects the Z2 grading and satisfies [x...

Along with additivity, this property is assumed in the definition of an antilinear map. It is also assumed that one of the two coordinates of a sesquilinear...

equation are actually nonlinear and pathological. Antilinear map – Conjugate homogeneous additive map Homogeneous function – Function with a multiplicative...

(literally, "half linear") may refer to: Antilinear map, also called a "semilinear map" Semilinear order Semilinear map Semilinear set Semilinearity (operator...

function f : H → C {\displaystyle \mathbb {C} } is called semilinear or antilinear if for all x, y ∈ H and all scalars c , Additive: f (x + y) = f (x) +...

product (no matter which argument is antilinear and no matter if it is real or complex) is a symmetric bilinear map that for any x , y ∈ H {\displaystyle...

complex vector space V with an invariant real structure, i.e., an antilinear equivariant map j : V → V {\displaystyle j\colon V\to V} which satisfies j 2 =...

}}={\text{Hom}}_{\overline {\mathbb {C} }}(V,\mathbb {C} )} of complex antilinear maps, is isomorphic to the real dual vector space Hom R ( V , R ) {\displaystyle...

map V × V → C that is linear in one argument and "twists" the linearity of the other argument by complex conjugation (referred to as being antilinear...

a Hilbert space, there is an antilinear isomorphism iV from V onto its continuous dual V′. For every bounded linear map T on V, the transpose and the...

⋅ | ⋅ ⟩ {\displaystyle \langle \,\cdot \,|\,\cdot \,\rangle } that is antilinear in its first coordinate (and linear in the second) then X R {\displaystyle...

g}}={\overline {g}}\star {\overline {f}},} The complex conjugate is an antilinear antiautomorphism. Note that, if one wishes to take functions valued in...

conjugate representation because the Hilbert space norm gives an antilinear bijective map from the representation to its dual representation. Self-dual complex...

{\displaystyle \phi } is faithful. We can define a (not necessarily bounded) antilinear operator S0 on H with dense domain MΩ by setting S 0 ( m Ω ) = m ∗ Ω {\displaystyle...