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.
Kolmogorov had shown in the previous year that any function of several variables can be constructed with a finite number of three-variable functions. Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering Hilbert's question when posed for the class of continuous functions.
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 vector space of all continuous antilinear functions on H is called the anti-dual space or complex conjugate dual space of H and is denoted by ¯ ′ (in contrast, the continuous dual space of H is denoted by ′), which we make into a normed space by endowing it with the canonical norm (defined in the same way as the canonical norm on the ...
For instance, given an isolated quantum mechanical system, with Hilbert space of states H, time evolution is a strongly continuous one-parameter unitary group on . The infinitesimal generator of this group is the system Hamiltonian
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication.
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.
In the case where the space is a space of functions, the functional is a "function of a function", [6] and some older authors actually define the term "functional" to mean "function of a function". However, the fact that X {\displaystyle X} is a space of functions is not mathematically essential, so this older definition is no longer prevalent.