What is the condition of separability of Hilbert space?

In quantum mechanics, the state space is a separable complex Hilbert space. A Hilbert space is separable if and only if it has a countable orthonormal basis [1, 2].

Why do we need 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.

What is Hilbert space in linear 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.

What are the properties of Hilbert space?

Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects of Hilbert space theory.

Are Hilbert spaces topological vector spaces?

As a complete normed space, Hilbert spaces are by definition also Banach spaces. As such they are topological vector spaces, in which topological notions like the openness and closedness of subsets are well defined.

What is the Hilbert space of the dot product?

Every finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ||x||, and to the angle θ between two vectors x and y by means of the formula

What is the difference between Hilbert space and metric space?

With a distance function defined in this way, any inner product space is a metric space, and sometimes is known as a pre-Hilbert space. Any pre-Hilbert space that is additionally also a complete space is a Hilbert space.

What is an orthonormal basis in a Hilbert space?

In a Hilbert space H, an orthonormal basis is a family {e k} k ∈ B of elements of H satisfying the conditions: Orthogonality: Every two different elements of B are orthogonal: ⟨e k, e j⟩ = 0 for all k, j ∈ B with k ≠ j. Normalization: Every element of the family has norm 1: ||e k|| = 1 for all k ∈ B.