What is Hilbert space theory?
In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. Hilbert spaces serve to clarify and generalize the concept of Fourier expansion and certain linear transformations such as the Fourier transform.
What is Hilbert space simple explanation?
A Hilbert space is a mathematical concept covering the extra-dimensional use of Euclidean space—i.e., a space with more than three dimensions. A Hilbert space uses the mathematics of two and three dimensions to try and describe what happens in greater than three dimensions. It is named after David Hilbert.
What is inner product in Hilbert space?
3.1-1 Definition (Inner product space, Hllbert space). An inner prod- uct space (or pre-Hilbert space) is a vector space X with an inner. product defined on X. A Hilbert space is a complete inner product. space (complete in the metric defined by the inner product; d. (
Is Hilbert space an inner product space?
A Hilbert space is a vector space equipped with an inner product which defines a distance function for which it is a complete metric space.
What is Hilbert space in quantum computing?
Hilbert Spaces are one of the most important mathematical constructs in quantum mechanics and quantum computation. A Hilbert space can be thought of as the state space in which all quantum state vectors “live”.
Who introduced Hilbert space?
Hilbert space, in mathematics, an example of an infinite-dimensional space that had a major impact in analysis and topology. The German mathematician David Hilbert first described this space in his work on integral equations and Fourier series, which occupied his attention during the period 1902–12.
Why do we need inner product?
Inner products are used to help better understand vector spaces of infinite dimension and to add structure to vector spaces. Inner products are often related to a notion of “distance” within the space, due to their positive-definite property.
How a Hilbert space is reflexive?
Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces. Reflexive Banach spaces are often characterized by their geometric properties.
What is product space in marketing?
The Product Space is a network representation of the relatedness or proximity between products traded in the global market.
What is the tensor product of Hilbert spaces?
In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product.
Is there a topology of the Hilbert space?
Since Hilbert spaces have inner products, one would like to introduce an inner product, and therefore a topology, on the tensor product that arise naturally from those of the factors. Let H1 and H2 be two Hilbert spaces with inner products , respectively.
What is the value of |Z| in a Hilbert space?
Hilbert spaces are often taken over the complex numbers. The complex plane denoted by C is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square root of the product of z with its complex conjugate : | z | 2 = z z ¯ . {\\displaystyle |z|^ {2}=z {\\overline {z}}\\,.}
Is the Hilbert space a symmetric monoidal category?
It is essentially the same universal property shared by all definitions of tensor products, irrespective of the spaces being tensored: this implies that any space with a tensor product is a symmetric monoidal category, and Hilbert spaces are a particular example thereof. . i = 1 , 2. {\\displaystyle i=1,2.}