What is special about Hilbert space?

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.

Is Hilbert space an algebra?

In mathematics, Hilbert spaces (named for David Hilbert) allow generalizing the methods of linear algebra and calculus from the two-dimensional and three dimensional Euclidean spaces to spaces that may have an infinite dimension.

Why is Hilbert space important in quantum mechanics?

In quantum mechanics, Hilbert space (a complete inner-product space) plays a central role in view of the interpretation associated with wave functions: absolute value of each wave function is interpreted as being a probability distribution function.

What is Hilbert space in physics?

A Hilbert space is a vector space equipped with an inner product operation, which allows defining a distance function and perpendicularity (known as orthogonality in this context). Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces.

Why do we use Hilbert space in quantum mechanics?

Are all Hilbert spaces Banach spaces?

Hilbert spaces with their norm given by the inner product are examples of Banach spaces. While a Hilbert space is always a Banach space, the converse need not hold. Therefore, it is possible for a Banach space not to have a norm given by an inner product.

Does Hilbert space exist?

Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane.