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The Hilbert projection theorem guarantees that this unique minimum point exists whenever is a non-empty closed and convex subset of a Hilbert space. However, such a minimum point can also exist in non-convex or non-closed subsets as well; for instance, just as long is C {\displaystyle C} is non-empty, if x ∈ C {\displaystyle x\in C} then min ...
In the Hilbert space view, this is the orthogonal projection of onto the kernel of the expectation operator, which a continuous linear functional on the Hilbert space (in fact, the inner product with the constant random variable 1), and so this kernel is a closed subspace.
A projection-valued measure π is homogeneous of multiplicity n if and only if the multiplicity function has constant value n. Clearly, Theorem. Any projection-valued measure π taking values in the projections of a separable Hilbert space is an orthogonal direct sum of homogeneous projection-valued measures:
The bounded operators on any Hilbert space form a von Neumann algebra, indeed a factor, of type I. If we have any unitary representation of a group G on a Hilbert space H then the bounded operators commuting with G form a von Neumann algebra G ′, whose projections correspond exactly to the closed subspaces of H invariant under G.
A square matrix is called a projection matrix if it is equal to its square, i.e. if =. [2]: p. 38 A square matrix is called an orthogonal projection matrix if = = for a real matrix, and respectively = = for a complex matrix, where denotes the transpose of and denotes the adjoint or Hermitian transpose of .
If a normal operator T on a finite-dimensional real [clarification needed] or complex Hilbert space (inner product space) H stabilizes a subspace V, then it also stabilizes its orthogonal complement V ⊥. (This statement is trivial in the case where T is self-adjoint.) Proof. Let P V be the orthogonal projection onto V.
In operator theory, a dilation of an operator T on a Hilbert space H is an operator on a larger Hilbert space K, whose restriction to H composed with the orthogonal projection onto H is T. More formally, let T be a bounded operator on some Hilbert space H, and H be a subspace of a larger Hilbert space H' . A bounded operator V on H' is a ...
In mathematics, the Hilbert metric, also known as the Hilbert projective metric, is an explicitly defined distance function on a bounded convex subset of the n-dimensional Euclidean space R n. It was introduced by David Hilbert ( 1895 ) as a generalization of Cayley's formula for the distance in the Cayley–Klein model of hyperbolic geometry ...