What is Hilbert space in quantum computing?

击 In quantum mechanics the state of a physical system is represented by a vector in a Hilbert space: a complex vector space with an inner product. ◦ The term “Hilbert space” is often reserved for an infinite-dimensional inner product space having the property that it is complete or closed.

What is the difference between a Banach space and a Hilbert space?

Similarly with normed spaces it will be easier to work with spaces where every Cauchy sequence is convergent. Such spaces are called Banach spaces and if the norm comes from an inner product then they are called Hilbert spaces. In other words the metric defined by the norm is complete.

What is Hilbert space PDF?

A Hilbert space is an inner product space (H,h·,·i) such that the induced Hilbertian norm is complete. Example 12.8. Let (X,M,µ) be a measure space then H := L2(X,M,µ) with. inner product. (f,g) = ZXf · ¯gdµ

Is a Hilbert space a manifold?

In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space.

Is Hilbert space infinite?

Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces.

Are all LP spaces Hilbert spaces?

However, neither Lp(R) nor ℓp is a Hilbert space when p = 2. Example 2.3 (Finite dimensional Hilbert spaces). The space Cn, finite-dimensional complex Euclidean space, is a Hilbert space.

Is a Hilbert space strictly convex?

Theorem 2.12. [1] Each Hilbert space, is uniformly convex.

Is a Hilbert space a subspace?

A subset M of Hilbert space H is a subspace of it is closed under the operation of forming linear combinations; i.e., for all x and y in M, C1x + C2y belongs to M for all scalars C1,C2.

Is a Hilbert space closed?

(b) Every finite dimensional subspace of a Hilbert space H is closed.

Is the Hilbert space a vector space?

A Hilbert space is a vector space with a positive definite inner product which is then used to define a norm and this in turn defines a metric making the Hilbert space into a metric topological vector space (the vector operations are continuous).

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 and Sobolev spaces?

Sobolev spaces, denoted by H s or W s, 2, are Hilbert spaces. These are a special kind of function space in which differentiation may be performed, but that (unlike other Banach spaces such as the Hölder spaces) support the structure of an inner product.

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.