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A Hilbert space is a vector space equipped with an inner product operation, which allows lengths and angles to be defined. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space.
More generally, Parseval's identity holds for arbitrary Hilbert spaces, not necessarily separable. When the Hilbert space is not separable any orthonormal basis is uncountable and we need to generalize the concept of a series to an unconditional sum as follows: let {} an orthonormal basis of a Hilbert space (where have arbitrary cardinality ...
The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
In mathematics, a singular trace is a trace on a space of linear operators of a separable Hilbert space that vanishes on operators of finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operators on a finite-dimensional ...
A von Neumann algebra that acts on a separable Hilbert space is called separable. Note that such algebras are rarely separable in the norm topology. The von Neumann algebra generated by a set of bounded operators on a Hilbert space is the smallest von Neumann algebra containing all those operators.
The problem of the differentiation of integrals is much harder in an infinite-dimensional setting. Consider a separable Hilbert space (H, , ) equipped with a Gaussian measure γ. As stated in the article on the Vitali covering theorem, the Vitali covering theorem fails for Gaussian measures on infinite-dimensional Hilbert spaces. Two results of ...
The Skorokhod integral operator which is conventionally denoted δ is defined as the adjoint of the Malliavin derivative in the white noise case when the Hilbert space is an space, thus for u in the domain of the operator which is a subset of ([,)), for F in the domain of the Malliavin derivative, we require
The simplest example of a direct integral are the L 2 spaces associated to a (σ-finite) countably additive measure μ on a measurable space X. Somewhat more generally one can consider a separable Hilbert space H and the space of square-integrable H-valued functions (,).