homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
(Hermitian form and Hermitian space)
Let be a real vector space equipped with a complex structure . Then a Hermitian form on is
a complex-valued real-bilinear form
such that this is symmetric sesquilinear, in that:
is complex-linear in the first argument;
for all
where denotes complex conjugation.
A Hermitian form is positive definite (often assumed by default) if for all
.
A complex vector space equipped with a (positive definite) Hermitian form is called a (positive definite) Hermitian space.
A positive-definite and complete Hermitian vector space is called a Hilbert space.
(basic properties of Hermitian forms)
Let be a positive definite Hermitian space (def. ). Then
the real part of the Hermitian form
is a Riemannian metric, hence a symmetric positive-definite real-bilinear form
the imaginary part of the Hermitian form
is a symplectic form, hence a non-degenerate skew-symmetric real-bilinear form
hence
The two components are related by
Finally
and so the Riemannian metrics on appearing from (and fully determining) Hermitian forms via are precisely those for which
These are called the Hermitian metrics.
The positive-definiteness of is immediate from that of . The symmetry of follows from the symmetric sesquilinearity of :
That is invariant under follows from its sesquilinarity
and this immediately implies the corresponding invariance of and .
Analogously it follows that is skew symmetric:
and the relation between the two components:
as well as
(relation between Kähler vector spaces and Hermitian spaces)
Given a real vector space with a linear complex structure , then the following are equivalent:
is a linear Kähler structure (def. );
is a Hermitian metric (2)
where and are related by (1)
While a non-degenerate inner product on a finite-dimensional real vector space is equivalently a linear isomorphism to its dual vector space
the analogous statement for Hermitian complex inner products fails, since the corresponding maps
are now complex anti-linear and hence not morphisms in the category of complex vector spaces.
What one does get is a complex-linear isomorphism to the anti-dual space?.
Another way to regard this situation is to observe that complex anti-linear involutions on non-degenerate Hermitian spaces are equivalently -module structures on the direct sum , regarded in the topos of -sets, for the ring object given by the complex numbers equipped with their involution by complex conjugation:
Notice that these -modules arising from (non-degenerate, finite-dimensional) Hermitian vector spaces this way happen to carry also a complex structure, hence a compatible module-structure by the actual complex numbers (i.e. equipped with the trivial involution), given by .
Using this, one may identify:
the space of linear operators on a Hermitian vector space as the equalizer of this imaginary rotation on the tensor square of the -module with the braiding and the identity,
among these that of hermitian operators as the further fixed locus of the involution action:
Among original articles:
See also:
and see the references at Hilbert space.
Last revised on October 21, 2022 at 07:32:08. See the history of this page for a list of all contributions to it.