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The Hilbertian tensor product of H 1 and H 2, sometimes denoted by H 1 ^ H 2, is the Hilbert space obtained by completing H 1 ⊗ H 2 for the metric associated to this inner product. [ 87 ] An example is provided by the Hilbert space L 2 ([0, 1]) .
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 } .
The norm induced by this inner product is the Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space). [4] The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with ...
Thus, unitary operators are just automorphisms of Hilbert spaces, i.e., they preserve the structure (the vector space structure, the inner product, and hence the topology) of the space on which they act. The group of all unitary operators from a given Hilbert space H to itself is sometimes referred to as the Hilbert group of H, denoted Hilb(H ...
The Hilbert space [,] is the space of ... is the space of the square-integrable functions on the interval [,] equipped with the inner product defined by , = (), (see ...
The Hilbert space has an associated inner product valued in 's underlying scalar field that is linear in one coordinate and antilinear in the other (as specified below). If H {\displaystyle H} is a complex Hilbert space ( F = C {\displaystyle \mathbb {F} =\mathbb {C} } ), then there is a crucial difference between the notations prevailing in ...
Hilbert C*-modules are mathematical objects that generalise the notion of Hilbert spaces (which are themselves generalisations of Euclidean space), in that they endow a linear space with an "inner product" that takes values in a C*-algebra.
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.