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An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. [2]
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 simplest example of a reproducing kernel Hilbert space is the space (,) where is a set and is the counting measure on . For x ∈ X {\displaystyle x\in X} , the reproducing kernel K x {\displaystyle K_{x}} is the indicator function of the one point set { x } ⊂ X {\displaystyle \{x\}\subset X} .
The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on H. They also form a Hilbert space, denoted by B HS (H) or B 2 (H), which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces, where H ∗ is the dual space of H.
Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category. [1]
The quotient space of by the vector subspace is an inner product space with the inner product defined by +, + := (),,, which is well-defined due to the Cauchy–Schwarz inequality. The Cauchy completion of A / I {\displaystyle A/I} in the norm induced by this inner product is a Hilbert space, which we denote by H {\displaystyle H} .
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.
The first three functions in the sequence () = on [,].As converges weakly to =.. The Hilbert space [,] is the space of the square-integrable functions on the interval [,] equipped with the inner product defined by