📂Hilbert Space

Reordering of Vector Spaces

Definition ¹

Let us suppose a sequence $\left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}}$ in a vector space $V$ is given. For a given bijection $\sigma : \mathbb{N} \to \mathbb{N}$, the following is called the reordering of $\left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}}$.

$$\left\{ \mathbf{v}_{\sigma (k) } \right\}_{k \in \mathbb{N}} = \left\{ \mathbf{v}_{\sigma (1)} , \mathbf{v}_{\sigma (2)} , \cdots \right\}$$

Explanation

Reordering is also called a Permutation, and as you can see, it is not a difficult concept but merely changing the order. In vector spaces, addition normally satisfies the commutative law, but mentioning such a definition is necessary because there is no guarantee that this property can be comfortably used for infinite series as well.

$$\mathbf{v} = \sum_{k \in \mathbb{N}} \left\langle \mathbf{v} , \mathbf{e}_{\sigma (k)} \right\rangle \mathbf{e}_{\sigma (k)}$$

In Hilbert space $H$, such series expansions are said to converge unconditionally when they hold for all $\mathbf{v} \in H$ regardless of the order, i.e., $\mathbf{e}_{k}$. Fortunately, we know that the independence of the orthonormal basis of Hilbert space does not depend on the order. Thus, we can consider the following theorem.

Theorem

If $\left\{ \mathbf{e}_{k} \right\}_{k \in \mathbb{N}}$ is the orthonormal basis of Hilbert space $H$, then for all $\mathbf{v} \in H$,

$$\mathbf{v} = \sum_{k \in \mathbb{N}} \left\langle \mathbf{v} , \mathbf{e}_{k} \right\rangle \mathbf{e}_{k}$$

unconditionally converges.

Proof

The independence of the orthonormal basis does not depend on the order.

Equivalence condition of orthonormal basis: Let us assume $H$ is a Hilbert space. For the orthonormal system $\left\{ \mathbf{e}_{k} \right\}_{k \in \mathbb{N}} \subset H$ of $H$, the following are equivalent:

• (i): $\left\{ \mathbf{e}_{k} \right\}_{k \in \mathbb{N}} \subset H$ is the orthonormal basis of $H$.
• (ii): For all $\mathbf{x}\in H$, $$\mathbf{x}= \sum_{k \in \mathbb{N}} \langle \mathbf{x}, \mathbf{e}_{k} \rangle \mathbf{e}_{k}$$

Since $\left\{ \mathbf{e}_{k} \right\}_{k \in \mathbb{N}}$ is the orthonormal basis of Hilbert space $H$ for all $\mathbf{v} \in H$,

$$\mathbf{v} = \sum_{k \in \mathbb{N}} \left\langle \mathbf{v} , \mathbf{e}_{k} \right\rangle \mathbf{e}_{k}$$

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