Is Hilbert Space Reflexive?

Hilbert spaces are prominent examples of reflexive Banach spaces .

What is a basis in Hilbert space?

Given a pre-Hilbert space H, an orthonormal basis for H is an orthonormal set of vectors with the property that every vector in H can be written as an infinite linear combination of the vectors in the basis . In this case, the orthonormal basis is sometimes called a Hilbert basis for H.

Does every Hilbert space have a basis?

Is every Hilbert space complete?

We will always use the norm defined in (6.1) on an inner product space. Definition 6.2 A Hilbert space is a complete inner product space . In particular, every Hilbert space is a Banach space with respect to the norm in (6.1).

Are the real numbers a Hilbert space?

Definition. A Hilbert space H is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product. A real inner product space is defined in the same way, except that H is a real vector space and the inner product takes real values.

Why is L1 not reflexive?

L1 (Rn) is not reflexive , so L∞(Rn) is not reflexive. This differs from the spaces Lp for 1 <p< ∞, which are reflexive. ... Recall: Let B be a separable Banach space, and let ξn ∈ B∗ be a such that ξn ≤ C. Then there exists a subsequence (ξnk ) which converges in σ(B∗,B).

Is c0 reflexive?

Hilbert spaces are prominent examples of reflexive Banach spaces . Reflexive Banach spaces are often characterized by their geometric properties.

Is a Hilbert space closed?

The subspace M is said to be closed if it contains all its limit points ; i.e., every sequence of elements of M that is Cauchy for the H-norm, converges to an element of M. ... (b) Every finite dimensional subspace of a Hilbert space H is closed.

How do you think about Hilbert space?

A Hilbert space H is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product. A real inner product space is defined in the same way, except that H is a real vector space and the inner product takes real values.

Is Hilbert space infinite?

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.

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 .

Why Hilbert space is infinite dimensional?

Hence, we have a need for infinite dimension Hilbert space to model quantum mechanics... It is because it operates with continuous wave functions, David . ... This will be one of the basis vectors of this Hilbert space. But in between measurements, the particle’s state is a combination of all possible measurement values.

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].

Which space is reflexive?

A space X is reflexive if and only if the space X∗ is reflexive. Another criteria of reflexivity of a Banach space X is weak compactness (cf. Weak topology) of the unit ball of this space. A reflexive space is weakly complete and a closed subspace of a reflexive space is reflexive.

What is the dual of l1?

(b) The dual of l1 is l∞ .

Are LP spaces reflexive?

Suppose (Ω, A,μ) is a σ -finite measure space. Let us prove that Lp = Lp(Ω,μ) is reflexive provided 1 <p< ∞ .