Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences

Let $\mathbb{R}$ be the real numbers and $\mathbb{N}$ the natural numbers; sequences are indexed by $\mathbb{N}$ as in Sequence in a Set, and limits of real sequences are those of Limit of a Sequence of Real Numbers. Let $V$ be a real vector space and let $\beta:V\times V\to\mathbb{R}$ be a map that is symmetric ($\beta(u,v)=\beta(v,u)$), bilinear ($v\mapsto\beta(u,v)$ is linear for each $u$) and positive semidefinite ($\beta(v,v)\ge0$), for all $u,v\in V$. For $v\in V$ let $\|v\|_{\beta}=\sqrt{\beta(v,v)}$, the nonnegative square root.

1. (Cauchy-Schwarz and the seminorm)¶ For all $u,v\in V$ and $c\in\mathbb{R}$:

2. (Cauchy sequences)¶ A sequence $(v_{k})_{k\in\mathbb{N}}$ in $V$ is $\beta$-Cauchy if for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $\|v_{k}-v_{l}\|_{\beta}<\varepsilon$ for all $k,l\ge N$. Let $C_{\beta}$ be the set of $\beta$-Cauchy sequences. Every constant sequence belongs to $C_{\beta}$; termwise sums and termwise real multiples of elements of $C_{\beta}$ belong to $C_{\beta}$; and with these termwise operations $C_{\beta}$ is a real vector space whose zero vector is the constant sequence $0$.

3. (Pairings)¶ For $u=(u_{k})$ and $v=(v_{k})$ in $C_{\beta}$ the real sequence $\bigl(\beta(u_{k},v_{k})\bigr)_{k\in\mathbb{N}}$ converges; write $\widehat\beta(u,v)$ for its limit. The map $\widehat\beta:C_{\beta}\times C_{\beta}\to\mathbb{R}$ is symmetric, bilinear and positive semidefinite, and $\widehat\beta(u,v)=\beta(u_{1},v_{1})$ when $u$ and $v$ are constant sequences.

4. (Null sequences)¶ Let $N_{\beta}$ be the set of $u=(u_{k})\in C_{\beta}$ such that the real sequence $(\|u_{k}\|_{\beta})_{k\in\mathbb{N}}$ converges to $0$. Then $N_{\beta}$ is a linear subspace of $C_{\beta}$; for $u\in C_{\beta}$ one has $u\in N_{\beta}$ if and only if $\widehat\beta(u,u)=0$; and $\widehat\beta(u,z)=0$ for all $u\in C_{\beta}$ and $z\in N_{\beta}$.

5. (Cosets)¶ For $u\in C_{\beta}$ let $[u]=\{u+z:\ z\in N_{\beta}\}$. For $u,v\in C_{\beta}$, $[u]=[v]$ holds if and only if $u-v\in N_{\beta}$. Consequently, if $[u]=[u']$ and $[v]=[v']$, then $[u+v]=[u'+v']$, $[cu]=[cu']$ for every $c\in\mathbb{R}$, and $\widehat\beta(u,v)=\widehat\beta(u',v')$.