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The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of all square-integrable functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of spinors.
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
The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space. The Hilbert polynomial and Hilbert series are important in computational algebraic geometry, as they are the easiest known way for computing the dimension and the degree of an algebraic variety defined by explicit ...
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 vector space of all continuous antilinear functions on H is called the anti-dual space or complex conjugate dual space of H and is denoted by ¯ ′ (in contrast, the continuous dual space of H is denoted by ′), which we make into a normed space by endowing it with the canonical norm (defined in the same way as the canonical norm on the ...
Therefore, the definition of rigged Hilbert space is in terms of a sandwich: . The most significant examples are those for which Φ is a nuclear space; this comment is an abstract expression of the idea that Φ consists of test functions and Φ* of the corresponding distributions.
The Hilbert transform can be understood in terms of a pair of functions f(x) and g(x) such that the function = + is the boundary value of a holomorphic function F(z) in the upper half-plane. [32] Under these circumstances, if f and g are sufficiently integrable, then one is the Hilbert transform of the other.
In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm.
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