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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.
In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory.
This definition applies to a Banach space, but of course other types of space exist as well; for example, topological vector spaces include Banach spaces, but can be more general. [12] [13] On the other hand, Banach spaces include Hilbert spaces, and it is these spaces that find the greatest application and the richest theoretical results. [14]
Unlike for Hilbert spaces, one may take infinite tensor products of von Neumann algebras, and for that matter C*-algebras of operators, without defining reference states. [3] This is one advantage of the "algebraic" method in quantum statistical mechanics.
Let be an arbitrary set and a Hilbert space of real-valued functions on , equipped with pointwise addition and pointwise scalar multiplication.The evaluation functional over the Hilbert space of functions is a linear functional that evaluates each function at a point ,
Fundamental theorem of Hilbert spaces: [1] Suppose that (H, B) is a Hausdorff pre-Hilbert space where B : H × H → is a sesquilinear form that is linear in its first coordinate and antilinear in its second coordinate.
As any Hilbert space, every space is linearly isometric to a suitable (), where the cardinality of the set is the cardinality of an arbitrary basis for this particular . If we use complex-valued functions, the space L ∞ {\displaystyle L^{\infty }} is a commutative C*-algebra with pointwise multiplication and conjugation.
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), then we say that converges unconditionally if for every > there exists a finite subset such that ...