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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Ole Christensen, Functions, Spaces, and Expansions: Mathematical Tools in Physics and Engineering (2010), p81 ↩︎