Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, and for $s\in\mathbb{R}$ let $|s|$ be its absolute value. Let $E$ be a real inner product space with inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$, distance $d$ and zero vector $0_{E}$, and let $\mathbb{N}$ be the set of natural numbers. Let $\mathrm{Sym}(E)$ be the set of bounded symmetric bilinear forms on $E$, with its norm $\lVert\cdot\rVert$, its order $\preceq$, and the operations $b_{1}+b_{2}$, $\lambda b$, the identity form $I=I_{E}$ and its multiples $cI$; let $0_{\mathrm{Sym}}$ denote the form with $0_{\mathrm{Sym}}(x,y)=0$ for all $x,y\in E$. Then the following hold, for all $b,b_{1},b_{2},b_{3},b_{1}',b_{2}'\in\mathrm{Sym}(E)$, all $x,y\in E$ and all $\lambda\in\mathbb{R}$.

1. (Vector space)(i) $0_{\mathrm{Sym}}\in\mathrm{Sym}(E)$, and $\mathrm{Sym}(E)$ with the operations $b_{1}+b_{2}$ and $\lambda b$ is a vector space over $\mathbb{R}$ with zero vector $0_{\mathrm{Sym}}$.

2. (Norm bound)(ii) $|b(x,y)|\le\lVert b\rVert\,|x|\,|y|$; and if $C$ is a real number with $0\le C$ and $|b(x',y')|\le C|x'||y'|$ for all $x',y'\in E$, then $\lVert b\rVert\le C$.

3. (Norm axioms)(iii) $\lVert b_{1}+b_{2}\rVert\le\lVert b_{1}\rVert+\lVert b_{2}\rVert$, $\lVert\lambda b\rVert=|\lambda|\,\lVert b\rVert$, and $\lVert b\rVert=0$ if and only if $b=0_{\mathrm{Sym}}$; moreover $\lVert I\rVert=1$ whenever $E\ne\{0_{E}\}$.

4. (Metric)(iv) The map $d_{\mathrm{Sym}}(b_{1},b_{2})=\lVert b_{1}-b_{2}\rVert$ is a metric on $\mathrm{Sym}(E)$.

5. (The quadratic form controls the norm)(v) For a real number $c$ with $0\le c$: $|b(x',x')|\le c\,|x'|^{2}$ for every $x'\in E$ if and only if $\lVert b\rVert\le c$.

6. (Order and norm)(vi) For a real number $c$ with $0\le c$: $-cI\preceq b\preceq cI$ if and only if $\lVert b\rVert\le c$. In particular $-\lVert b\rVert I\preceq b\preceq\lVert b\rVert I$.

7. (Partial order)(vii) $b\preceq b$; if $b_{1}\preceq b_{2}$ and $b_{2}\preceq b_{3}$ then $b_{1}\preceq b_{3}$; and if $b_{1}\preceq b_{2}$ and $b_{2}\preceq b_{1}$ then $b_{1}=b_{2}$.

8. (Compatibility of the order with the operations)(viii) If $b_{1}\preceq b_{2}$, then $b_{1}+b\preceq b_{2}+b$, $\lambda b_{1}\preceq\lambda b_{2}$ whenever $0\le\lambda$, and $\lambda b_{2}\preceq\lambda b_{1}$ whenever $\lambda\le 0$. If $b_{1}\preceq b_{2}$ and $b_{1}'\preceq b_{2}'$, then $b_{1}+b_{1}'\preceq b_{2}+b_{2}'$.

9. (Continuity)(ix) $|b(x,x)-b(y,y)|\le\lVert b\rVert\,(|x|+|y|)\,|x-y|$, $|b_{1}(x,y)-b_{2}(x,y)|\le\lVert b_{1}-b_{2}\rVert\,|x|\,|y|$, and for all $x',y'\in E$, $|b(x',y')-b(x,y)|\le\lVert b\rVert\bigl(|x'-x|\,|y'|+|x|\,|y'-y|\bigr)$. Consequently, if $(x_{m})_{m\in\mathbb{N}}$ and $(y_{m})_{m\in\mathbb{N}}$ are sequences in $E$ converging to $x$ and to $y$ in $(E,d)$, and $(b_{m})_{m\in\mathbb{N}}$ is a sequence in $\mathrm{Sym}(E)$ converging to $b$ in $(\mathrm{Sym}(E),d_{\mathrm{Sym}})$, then the sequence of real numbers $(b_{m}(x_{m},y_{m}))$ converges to $b(x,y)$.

10. (Restriction)(x) Let $V$, $\langle\cdot,\cdot\rangle_{V}$, $|\cdot|_{V}$ and $b|_{V}$ be as in Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (v), and write $\lVert\cdot\rVert_{V}$, $\preceq_{V}$ and $I_{V}$ for the norm, order and identity form of $\mathrm{Sym}(V)$. Then $\lVert b|_{V}\rVert_{V}\le\lVert b\rVert$; $(b_{1}+b_{2})|_{V}=b_{1}|_{V}+b_{2}|_{V}$ and $(\lambda b)|_{V}=\lambda\,(b|_{V})$; $b_{1}\preceq b_{2}$ implies $b_{1}|_{V}\preceq_{V}b_{2}|_{V}$; and $I_{E}|_{V}\preceq_{V}I_{V}$.