Complete orthonormal sequence

1. What is complete orthonormal sequence?
2. Is a Hilbert space complete?
3. Is a Hilbert space closed?
4. Is RN a Hilbert space?

What is complete orthonormal sequence?

Definition ‍14 An orthonormal sequence (e_i) in a Hilbert space H is complete if the identities ⟨ y,e_k ⟩=0 for all k imply y=0. A complete orthonormal sequence is also called orthonormal basis in H. Theorem ‍15 ‍(on Orthonormal Basis) Let e_i be an orthonormal basis in a Hilber space H. Then for any x∈ H we have. x=

Is a Hilbert space complete?

Thus, any inner product space is a normed linear space. 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.

Is a Hilbert space closed?

(b) Every finite dimensional subspace of a Hilbert space H is closed. For example, if M denotes the span of finitely many elements x1, ... . xN in H, then the set M of all possible linear combinations of these elements is finite dimensional (of dimension N), hence it is closed in H.

Is RN a Hilbert space?

For example, Rn is a Hilbert space under the usual dot product: 〈v,w〉 = v · w = v1w1 + ··· + vnwn. More generally, a finite-dimensional inner product space is a Hilbert space. The following theorem provides examples of infinite-dimensional Hilbert spaces.