Proof of The Bounded Symmetric Operator Represented by a Bounded Symmetric Bilinear Form on a Real Hilbert Space

Throughout we use that $0\le\lVert b\rVert$ for every $b\in\mathrm{Sym}(H)$ and $0\le\lVert T\rVert$ for every $T\in\mathcal{L}(H)$: by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (ii) and Bounded Linear Maps and Bounded Linear Functionals on Real Inner Product Spaces, and the Operator Norm (ii) these numbers are greatest lower bounds of sets of nonnegative real numbers, of which $0$ is therefore a lower bound. We also use the symmetry $\langle x,y\rangle=\langle y,x\rangle$ of Real Inner Product Spaces (i) without comment.

Claim 1. Let $b\in\mathrm{Sym}(H)$ and fix $x\in H$. The map $\ell_{x}:H\to\mathbb{R}$ given by $\ell_{x}(y)=b(x,y)$ is a linear functional: for $y,y'\in H$ and $\lambda\in\mathbb{R}$, the symmetry, additivity and homogeneity of Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (i) give

$$b(x,y+y')=b(y+y',x)=b(y,x)+b(y',x)=b(x,y)+b(x,y'),\qquad b(x,\lambda y)=b(\lambda y,x)=\lambda\,b(y,x)=\lambda\,b(x,y).$$

It is bounded, since $|b(x,y)|\le\lVert b\rVert\,|x|\,|y|$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (ii), so that the constant $\lVert b\rVert\,|x|$ serves in Bounded Linear Maps and Bounded Linear Functionals on Real Inner Product Spaces, and the Operator Norm (iii). By The Riesz Representation Theorem for a Real Hilbert Space (i) there is $z\in H$ with $b(x,y)=\langle y,z\rangle=\langle z,y\rangle$ for every $y\in H$, and by The Riesz Representation Theorem for a Real Hilbert Space (ii) such a $z$ is unique. Setting $T_{b}x=z$ for each $x\in H$ defines a map $T_{b}:H\to H$ with $b(x,y)=\langle T_{b}x,y\rangle$ for all $x,y\in H$. If $T:H\to H$ is any map with this property, then for each $x$ we have $\langle y,Tx\rangle=\langle y,T_{b}x\rangle$ for every $y\in H$, so $Tx=T_{b}x$ by The Riesz Representation Theorem for a Real Hilbert Space (ii); the map is therefore unique.

The map $T_{b}$ is linear: for $x,x',y\in H$ and $\lambda\in\mathbb{R}$, Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (i) and conditions (b) and (c) of Real Inner Product Spaces (i) give

$$\langle T_{b}(x+x'),y\rangle=b(x+x',y)=b(x,y)+b(x',y)=\langle T_{b}x+T_{b}x',y\rangle,\qquad\langle T_{b}(\lambda x),y\rangle=b(\lambda x,y)=\lambda\,b(x,y)=\langle\lambda\,T_{b}x,y\rangle ,$$

for every $y\in H$, whence $T_{b}(x+x')=T_{b}x+T_{b}x'$ and $T_{b}(\lambda x)=\lambda\,T_{b}x$ by The Riesz Representation Theorem for a Real Hilbert Space (ii). It is bounded: for $x\in H$,

$$|T_{b}x|^{2}=\langle T_{b}x,T_{b}x\rangle=b(x,T_{b}x)\le\bigl|b(x,T_{b}x)\bigr|\le\lVert b\rVert\,|x|\,|T_{b}x| ,$$

by Real Inner Product Spaces (ii), claim 3 of Properties of the Absolute Value in an Ordered Field and Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (ii). If $T_{b}x=0_{H}$ then $|T_{b}x|=0\le\lVert b\rVert\,|x|$, the last inequality by claim 5 of Elementary Arithmetic in an Ordered Field. Otherwise $|T_{b}x|$ is positive by Elementary Identities in a Real Inner Product Space (iii), and multiplying the displayed inequality by the nonnegative number $|T_{b}x|^{-1}$ (claim 5 of Elementary Arithmetic in an Ordered Field, with claim 7 of Elementary Order Arithmetic in an Ordered Field for its positivity) gives $|T_{b}x|\le\lVert b\rVert\,|x|$. In either case $T_{b}\in\mathcal{L}(H)$ by Bounded Linear Maps and Bounded Linear Functionals on Real Inner Product Spaces, and the Operator Norm (i).

Claim 2. For $x,y\in H$, using claim 1 and the symmetry of Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (i),

$$\langle T_{b}x,y\rangle=b(x,y)=b(y,x)=\langle T_{b}y,x\rangle=\langle x,T_{b}y\rangle .$$

Claim 3. The bound $|T_{b}x|\le\lVert b\rVert\,|x|$ established in claim 1 holds for every $x\in H$, and $0\le\lVert b\rVert$, so Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces (i) gives $\lVert T_{b}\rVert\le\lVert b\rVert$. Conversely, for all $x,y\in H$, The Cauchy–Schwarz Inequality in a Real Inner Product Space, Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces (i) and claim 5 of Elementary Arithmetic in an Ordered Field give

$$\bigl|b(x,y)\bigr|=\bigl|\langle T_{b}x,y\rangle\bigr|\le|T_{b}x|\,|y|\le\lVert T_{b}\rVert\,|x|\,|y| ,$$

so $\lVert b\rVert\le\lVert T_{b}\rVert$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (ii), since $0\le\lVert T_{b}\rVert$. The order of $\mathbb{R}$ is antisymmetric, being a total order, so $\lVert T_{b}\rVert=\lVert b\rVert$.

Claim 4. By Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iv) and condition (b) of Real Inner Product Spaces (i), for all $x,y\in H$,

$$\langle T_{b_{1}+b_{2}}x,y\rangle=(b_{1}+b_{2})(x,y)=b_{1}(x,y)+b_{2}(x,y)=\langle T_{b_{1}}x,y\rangle+\langle T_{b_{2}}x,y\rangle=\langle T_{b_{1}}x+T_{b_{2}}x,y\rangle .$$

Since $T_{b_{1}}+T_{b_{2}}$ is a map from $H$ to $H$ representing $b_{1}+b_{2}$ in the sense of claim 1, the uniqueness there gives $T_{b_{1}+b_{2}}=T_{b_{1}}+T_{b_{2}}$. The same argument, with condition (c) of Real Inner Product Spaces (i) in place of (b), gives $T_{\lambda b_{1}}=\lambda\,T_{b_{1}}$. Furthermore $I(x,y)=\langle x,y\rangle=\langle\mathrm{id}_{H}x,y\rangle$ and $(cI)(x,y)=c\,\langle x,y\rangle=\langle c\,x,y\rangle$ by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iv) and condition (c), while $0_{\mathrm{Sym}}(x,y)=0=\langle 0_{H},y\rangle$ by Elementary Identities in a Real Inner Product Space (ii); the uniqueness in claim 1 gives $T_{I}=\mathrm{id}_{H}$, $T_{cI}x=c\,x$ and $T_{0_{\mathrm{Sym}}}x=0_{H}$ for every $x\in H$.

Claim 5. The map $b_{T}$ is symmetric, since $b_{T}(x,y)=\langle Tx,y\rangle=\langle x,Ty\rangle=\langle Ty,x\rangle=b_{T}(y,x)$ by hypothesis. It is additive and homogeneous in its first argument, because $T$ is a linear map and the inner product satisfies conditions (b) and (c) of Real Inner Product Spaces (i). It is bounded, since The Cauchy–Schwarz Inequality in a Real Inner Product Space, Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces (i) and claim 5 of Elementary Arithmetic in an Ordered Field give $|\langle Tx,y\rangle|\le|Tx|\,|y|\le\lVert T\rVert\,|x|\,|y|$ for all $x,y\in H$. Hence $b_{T}\in\mathrm{Sym}(H)$ by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (i). Since $b_{T}(x,y)=\langle Tx,y\rangle$ for all $x,y\in H$, the map $T$ represents $b_{T}$ in the sense of claim 1, so $T_{b_{T}}=T$ by the uniqueness there.