Search results
Results from the WOW.Com Content Network
Let and be Hilbert spaces, and let : be an unbounded operator from into . Suppose that is a closed operator and that is densely defined, that is, is dense in . Let : denote the adjoint of .
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. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz.
A rigged Hilbert space is a pair (H, Φ) with H a Hilbert space, Φ a dense subspace, such that Φ is given a topological vector space structure for which the inclusion map:, is continuous. [ 4 ] [ 5 ] Identifying H with its dual space H * , the adjoint to i is the map i ∗ : H = H ∗ → Φ ∗ . {\displaystyle i^{*}:H=H^{*}\to \Phi ^{*}.}
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
Theorem — For each non-zero real number h there is an irreducible representation U h acting on the Hilbert space L 2 (R n) by [((,,))] = (+) (+). All these representations are unitarily inequivalent ; and any irreducible representation which is not trivial on the center of H n is unitarily equivalent to exactly one of these.
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.
Lemma — If A, B are bounded operators on a Hilbert space H, and A*A ≤ B*B, then there exists a contraction C such that A = CB. Furthermore, C is unique if Ker ( B* ) ⊂ Ker ( C ). The operator C can be defined by C ( Bh ) = Ah , extended by continuity to the closure of Ran ( B ), and by zero on the orthogonal complement of Ran( B ) .
A feature map is a map :, where is a Hilbert space which we will call the feature space. The first sections presented the connection between bounded/continuous evaluation functions, positive definite functions, and integral operators and in this section we provide another representation of the RKHS in terms of feature maps.