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Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz.
The Hilbert symbol can also be used to denote the central simple algebra over K with basis 1,i,j,k and multiplication rules =, =, = =.In this case the algebra represents an element of order 2 in the Brauer group of K, which is identified with -1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.
(The exact structure of this Hilbert space depends on the situation.) In bra–ket notation, for example, an electron might be in the "state" |ψ . (Technically, the quantum states are rays of vectors in the Hilbert space, as c |ψ corresponds to the same state for any nonzero complex number c.)
The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
The latter space is a Hilbert space and the Fourier transform is an isomorphism of Hilbert spaces. [ nb 10 ] Its basis is { e i , i ∈ Z } with e i ( j ) = δ ij , i , j ∈ Z . The most basic example of spanning polynomials is in the space of square integrable functions on the interval [–1, 1] for which the Legendre polynomials is a Hilbert ...
The above definition in the Hilbert space setting is really just an application of the Banach space case when one identifies a Hilbert space with its dual (via the Riesz representation theorem). Then it is only natural that we can also obtain the adjoint of an operator A : H → E {\displaystyle A:H\to E} , where H {\displaystyle H} is a ...
The resulting direct sum is a Hilbert space which contains the given Hilbert spaces as mutually orthogonal subspaces. If infinitely many Hilbert spaces H i {\displaystyle H_{i}} for i ∈ I {\displaystyle i\in I} are given, we can carry out the same construction; notice that when defining the inner product, only finitely many summands will be ...
In mathematics and control theory, H 2, or H-square is a Hardy space with square norm. It is a subspace of L 2 space, and is thus a Hilbert space. In particular, it is a reproducing kernel Hilbert space.