Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, let $E$ be a real inner product space with inner product $\langle\cdot,\cdot\rangle$ and norm $|\cdot|$, and for a real number $s$ let $|s|$ be its absolute value.

1. (Bounded symmetric bilinear forms)(i) A bounded symmetric bilinear form on $E$ is a map $b$ assigning to each pair $x,y$ of elements of $E$ a real number $b(x,y)$ such that for all $x,y,z\in E$ and all $\lambda\in\mathbb{R}$: $b(x,y)=b(y,x)$; $b(x+y,z)=b(x,z)+b(y,z)$; $b(\lambda x,y)=\lambda\,b(x,y)$; and there is a real number $C$ with $|b(x,y)|\le C\,|x|\,|y|$ for all $x,y\in E$. We write $\mathrm{Sym}(E)$ for the set of all bounded symmetric bilinear forms on $E$.

2. (Norm)(ii) Let $b\in\mathrm{Sym}(E)$ and let $B_{b}$ be the set of real numbers $C$ with $0\le C$ such that $|b(x,y)|\le C|x||y|$ for all $x,y\in E$. The set $B_{b}$ is nonempty: if $C$ is as in clause 1 and $0\le C$ then $C\in B_{b}$, while if $C<0$ then, by claim 5 of Elementary Arithmetic in an Ordered Field applied to $C\le 0$ with a nonnegative multiplier, $|b(x,y)|\le C|x||y|\le 0\le 0\cdot|x||y|$ for all $x,y$, so $0\in B_{b}$. It is bounded below by $0$, so by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below it has a greatest lower bound. The norm of $b$ is

$$\lVert b\rVert=\inf B_{b}.$$

3. (Order)(iii) For $b_{1},b_{2}\in\mathrm{Sym}(E)$ we write $b_{1}\preceq b_{2}$, or equivalently $b_{2}\succeq b_{1}$, if $b_{1}(x,x)\le b_{2}(x,x)$ for every $x\in E$.

4. (Sums, multiples and the identity form)(iv) For $b_{1},b_{2}\in\mathrm{Sym}(E)$ and $\lambda\in\mathbb{R}$ the maps $b_{1}+b_{2}$ and $\lambda b_{1}$ given by $(b_{1}+b_{2})(x,y)=b_{1}(x,y)+b_{2}(x,y)$ and $(\lambda b_{1})(x,y)=\lambda\,b_{1}(x,y)$ belong to $\mathrm{Sym}(E)$: symmetry, additivity and homogeneity hold termwise, and if $C_{1},C_{2}$ are constants as in clause 1 for $b_{1},b_{2}$, then $C_{1}+C_{2}$ serves for $b_{1}+b_{2}$ and $|\lambda|C_{1}$ serves for $\lambda b_{1}$: indeed $|b_{1}(x,y)+b_{2}(x,y)|\le|b_{1}(x,y)|+|b_{2}(x,y)|\le C_{1}|x||y|+C_{2}|x||y|=(C_{1}+C_{2})|x||y|$ by the triangle inequality (claim 5 of Properties of the Absolute Value in an Ordered Field), the compatibility of the order with addition (an axiom of Ordered Field) applied to each summand, and transitivity, and $|\lambda\,b_{1}(x,y)|=|\lambda|\,|b_{1}(x,y)|\le|\lambda|C_{1}|x||y|$ by multiplicativity of the absolute value (claim 4 of Properties of the Absolute Value in an Ordered Field) and claim 5 of Elementary Arithmetic in an Ordered Field, since $0\le|\lambda|$ by claim 1 of Properties of the Absolute Value in an Ordered Field. We write $b_{1}-b_{2}$ for $b_{1}+(-1)b_{2}$ and $-b_{1}$ for $(-1)b_{1}$. The inner product itself belongs to $\mathrm{Sym}(E)$: symmetry, additivity and homogeneity are conditions (a), (b), (c) of Real Inner Product Spaces (i), and $|\langle x,y\rangle|\le 1\cdot|x||y|$ by The Cauchy–Schwarz Inequality in a Real Inner Product Space. It is called the identity form and denoted $I_{E}$, or $I$ when $E$ is clear; for $c\in\mathbb{R}$ we write $cI$ for $c\,I_{E}$, so that $(cI)(x,y)=c\,\langle x,y\rangle$, and $-cI$ for $(-c)I$.

5. (Restriction to a subspace with a stronger inner product)(v) Let $V$ be a linear subspace of $E$, which is a vector space over $\mathbb{R}$ under the restricted operations of $E$ by claim 1 of A Linear Subspace is a Vector Space and Inherits an Inner Product, and suppose $V$ is equipped with an inner product $\langle\cdot,\cdot\rangle_{V}$ making it a real inner product space, with norm $|\cdot|_{V}$, such that $|x|\le|x|_{V}$ for every $x\in V$. For $b\in\mathrm{Sym}(E)$ the map $b|_{V}$ assigning $b(x,y)$ to each pair $x,y\in V$ belongs to $\mathrm{Sym}(V)$: the vector operations of $V$ are the restrictions of those of $E$, so symmetry, additivity and homogeneity are inherited, and if $C\in B_{b}$, a set nonempty by clause 2, so that $0\le C$, then $|b(x,y)|\le C|x||y|\le C|x|_{V}|y|\le C|x|_{V}|y|_{V}$ for all $x,y\in V$, by claim 5 of Elementary Arithmetic in an Ordered Field applied to $|x|\le|x|_{V}$ with the nonnegative multiplier $C|y|$ and then to $|y|\le|y|_{V}$ with the nonnegative multiplier $C|x|_{V}$. The form $b|_{V}$ is called the restriction of $b$ to $V$.