Search results
Results from the WOW.Com Content Network
Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space. A Hilbert space is a special case of a Banach space.
where H(D) is the space of holomorphic functions in D. Then L 2,h (D) is a Hilbert space: it is a closed linear subspace of L 2 (D), and therefore complete in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ƒ in D
The first three functions in the sequence () = on [,].As converges weakly to =.. The Hilbert space [,] is the space of the square-integrable functions on the interval [,] equipped with the inner product defined by
It is clear from the definition of the inner product on the GNS Hilbert space that the state can be recovered as a vector state on . This proves the theorem. This proves the theorem. The method used to produce a ∗ {\displaystyle *} -representation from a state of A {\displaystyle A} in the proof of the above theorem is called the GNS ...
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space [1] [2]) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar , often denoted with angle brackets such as in a , b {\displaystyle \langle a,b\rangle } .
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 ,
The name spectral theory was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid , in an infinite-dimensional setting.
The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space. The Hilbert polynomial and Hilbert series are important in computational algebraic geometry, as they are the easiest known way for computing the dimension and the degree of an algebraic variety defined by explicit ...