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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.
In the special case of Fourier series for the unit circle, the operators become the classical Cauchy transform, the orthogonal projection onto Hardy space, and the Hilbert transform a real orthogonal linear complex structure. In general the Cauchy transform is a non-self-adjoint idempotent and the Hilbert transform a non-orthogonal complex ...
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 .
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:
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.
Operators E in a von Neumann algebra for which E = EE = E* are called projections; they are exactly the operators which give an orthogonal projection of H onto some closed subspace. A subspace of the Hilbert space H is said to belong to the von Neumann algebra M if it is the image of some projection in M.
Hilbert projection theorem – On closed convex subsets in Hilbert space Orthogonal projection – Idempotent linear transformation from a vector space to itself Pages displaying short descriptions of redirect targets