In mathematics, in the field of functional analysis, an indefinite inner product space is an infinite-dimensional complex vector space
obeying The indefinite inner product space itself is not necessarily a Hilbert space; but the existence of a positive semi-definite inner product on
implies that one can form a quotient space on which there is a positive definite inner product.
In the theory of Krein spaces it is common to call such an hermitian form an indefinite inner product.
The following subsets are defined in terms of the square norm induced by the indefinite inner product: A subspace
) is called positive (negative) semi-definite, and a subspace lying within
Let our indefinite inner product space also be equipped with a decomposition into a pair of subspaces
, called the fundamental decomposition, which respects the complex structure on
is called a Krein space, subject to the existence of a majorant topology on
(a locally convex topology where the inner product is jointly continuous).
is called the (real phase) metric operator or fundamental symmetry, and may be used to define the Hilbert inner product
: On a Krein space, the Hilbert inner product is positive definite, giving
the structure of a Hilbert space (under a suitable topology).
are part of the neutral subspace of the Hilbert inner product, because an element
will have a positive square norm under the Hilbert inner product.
We note that the definition of the indefinite inner product as a Hermitian form implies that: (Note: This is not correct for complex-valued Hermitian forms.
is equal to the square norm of their average
has the wrong sign to be the square norm of
Similar arguments about the Hilbert inner product (which can be demonstrated to be a Hermitian form, therefore justifying the name "inner product") lead to the conclusion that its neutral space is precisely
It therefore induces a positive definite inner product (also denoted
Krein spaces arise naturally in situations where the indefinite inner product has an analytically useful property (such as Lorentz invariance) which the Hilbert inner product lacks.
It is also common for one of the two inner products, usually the indefinite one, to be globally defined on a manifold and the other to be coordinate-dependent and therefore defined only on a local section.
depends on the chosen fundamental decomposition, which is, in general, not unique.
Proposition 1.1 and 1.2 in the paper of H. Langer below) that any two metric operators
Although the Hilbert inner products on these quotient spaces do not generally coincide, they induce identical square norms, in the sense that the square norms of the equivalence classes
All topological notions in a Krein space, like continuity, closed-ness of sets, and the spectrum of an operator on
, are understood with respect to this Hilbert space topology.
If, in addition, this is a direct sum, we write
(Conventionally, the indefinite inner product is given the sign that makes
A symmetric operator A on an indefinite inner product space K with domain K is called a Pesonen operator if (x,x) = 0 = (x,Ax) implies x = 0.