Proof of The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It

This proof uses: the clauses Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §cauchy-space, Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §pairing, Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §null and Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §cosets, and the definition The Hilbert Completion of a Real Vector Space with a Positive Semidefinite Symmetric Bilinear Form; Vector Space over a Field, Linear Map and claims 1 and 2 of Elementary Identities in a Vector Space; Real Inner Product Space, The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity, Real Hilbert Space, Complete Metric Space, Convergent Sequence in a Metric Space, Cauchy Sequence in a Metric Space, Uniqueness of Limits in a Metric Space, Continuous Map Between Metric Spaces, Dense Subset of a Topological Space, Closure of a Subset of a Topological Space and Characterization of the Closure in a Metric Space by Open Balls; for real numbers Elementary Order Arithmetic in an Ordered Field, Elementary Arithmetic in an Ordered Field, Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, Existence and Uniqueness of the Nonnegative Square Root and claims 1, 3(a), 3(d), 3(e) and 5 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities; for real limits Limit of a Sequence of Real Numbers, Arithmetic of Limits of Real Sequences, Order Properties of Limits of Real Sequences and claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences; for indices, the maximum of finitely many natural numbers as in Step 3 of the proof of Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences (claim 1 of Elementary Properties of the Maximum of Two Elements and claim 1 of Properties of the Order on the Natural Numbers); and the axiom of countable choice, used once, in Step 6. Write $\|\cdot\|$ for $\|\cdot\|_{\beta}$, $H=H_{\beta}$, $J=J_{\beta}$, and $\bar v$ for the constant sequence with value $v\in V$; for $\varepsilon>0$ the numbers $\varepsilon/2$ and $\varepsilon/4=(\varepsilon/2)/2$ are positive with $\varepsilon/2+\varepsilon/2=\varepsilon$ and $\varepsilon/4+\varepsilon/4=\varepsilon/2$ (claim 8 of Elementary Order Arithmetic in an Ordered Field).

Step 1 (claim 1, vector space). The operations of The Hilbert Completion of a Real Vector Space with a Positive Semidefinite Symmetric Bilinear Form §completion are well defined by Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §cosets, and $C_{\beta}$ is a real vector space by Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §cauchy-space. Each of conditions 1, 2 and 5 to 8 of Vector Space over a Field for $H$ follows from the same condition in $C_{\beta}$ by writing every element as a coset; for instance $([u]+[v])+[w]=[(u+v)+w]=[u+(v+w)]=[u]+([v]+[w])$ and $c([u]+[v])=[c(u+v)]=[cu+cv]=c[u]+c[v]$. Condition 3 holds with $[\bar0]$, since $[u]+[\bar0]=[u+\bar0]=[u]$, and condition 4 with $[(-1)u]$, since $[u]+[(-1)u]=[u+(-1)u]=[\bar0]$. By claims 1 and 2 of Elementary Identities in a Vector Space, $[\bar0]$ is the zero vector of $H$, $-[u]=[(-1)u]$, and therefore $[u]-[v]=[u-v]$ for $u,v\in C_{\beta}$.

Step 2 (claim 1, inner product). We check conditions (a) to (d) of Real Inner Product Space §inner-product for $\langle[u],[v]\rangle_{H}=\widehat\beta(u,v)$, which is well defined by Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §cosets. By Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §pairing, $\widehat\beta$ is symmetric and linear in its second argument, hence by symmetry also in its first; so $\langle[u]+[v],[w]\rangle_{H}=\widehat\beta(u+v,w)=\widehat\beta(u,w)+\widehat\beta(v,w)$ and $\langle c[u],[w]\rangle_{H}=\widehat\beta(cu,w)=c\,\widehat\beta(u,w)$, which with symmetry gives (a) to (c). For (d), $0\le\widehat\beta(u,u)$ by Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §pairing; if $\widehat\beta(u,u)=0$, then $u\in N_{\beta}$ by Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §null, so $u-\bar0=u\in N_{\beta}$ and $[u]=[\bar0]$ by Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §cosets, the zero vector of $H$ (Step 1). So $H$ is a real inner product space. We record, for $u\in C_{\beta}$,

Indeed $\|u_{k}\|^{2}=\beta(u_{k},u_{k})\to\widehat\beta(u,u)=|[u]|^{2}$ by Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §pairing and Real Inner Product Space §norm; by claim 3(e) of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities, $\sqrt{\|u_{k}\|^{2}}\to\sqrt{|[u]|^{2}}$, and $\sqrt{s^{2}}=s$ for $s\ge0$ by claim 5 there, so $\|u_{k}\|\to|[u]|$.

Step 3 (claim 2). By The Hilbert Completion of a Real Vector Space with a Positive Semidefinite Symmetric Bilinear Form §canonical-map, $Jv=[\bar v]$. Since $\overline{v+w}=\bar v+\bar w$ and $\overline{cv}=c\,\bar v$ termwise, $J(v+w)=[\bar v+\bar w]=Jv+Jw$ and $J(cv)=c\,Jv$, so $J$ is linear (Linear Map). By the last assertion of Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §pairing, $\langle Jv,Jw\rangle_{H}=\widehat\beta(\bar v,\bar w)=\beta(v,w)$. In particular $|Jv|^{2}=\beta(v,v)=\|v\|^{2}$ with $|Jv|,\|v\|\ge0$, so $|Jv|=\|v\|$ by claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Finally $Jv=[\bar0]$ holds iff $\bar v=\bar v-\bar0\in N_{\beta}$ (Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §cosets) iff $\widehat\beta(\bar v,\bar v)=0$ (Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §null) iff $\beta(v,v)=0$.

Step 4 (claim 3, convergence). Let $u=(u_{k})\in C_{\beta}$ and $\varepsilon>0$, and choose $N$ with $\|u_{k}-u_{l}\|<\varepsilon/2$ for $k,l\ge N$. Fix $k\ge N$. By Step 1, $Ju_{k}-[u]=[\bar u_{k}-u]$, and $\bar u_{k}-u$ is the sequence $(u_{k}-u_{l})_{l\in\mathbb{N}}$, which lies in $C_{\beta}$; so by (N), $L_{k}:=|Ju_{k}-[u]|=\lim_{l\to\infty}\|u_{k}-u_{l}\|$. If $\varepsilon/2<L_{k}$, claim 3(d) of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities would give $N_{0}$ with $\|u_{k}-u_{l}\|>\varepsilon/2$ for all $l\ge N_{0}$, which fails at $l=\max\{N,N_{0}\}$ because $k,l\ge N$; hence $L_{k}\le\varepsilon/2$, the order of $\mathbb{R}$ being total, and $L_{k}<\varepsilon$ by claim 2 of Elementary Order Arithmetic in an Ordered Field. Thus $d(Ju_{k},[u])=|Ju_{k}-[u]|<\varepsilon$ for all $k\ge N$, where $d$ is the distance of $H$, a metric by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric; that is, $Ju_{k}\to[u]$ in the sense of Convergent Sequence in a Metric Space.

Step 5 (claim 3, density). By Closure of a Subset of a Topological Space, $\operatorname{cl}_{H}(J(V))\subseteq H$. Let $h\in H$, say $h=[u]$, and $\varepsilon>0$. By Step 4 there is $k$ with $d(Ju_{k},h)<\varepsilon$, so $d(h,Ju_{k})<\varepsilon$ with $Ju_{k}\in J(V)$ (symmetry of the metric, condition 3 of Metric Space). By the implication from condition 3 to condition 1 of Characterization of the Closure in a Metric Space by Open Balls, $h\in\operatorname{cl}_{H}(J(V))$. Hence $\operatorname{cl}_{H}(J(V))=H$, and $J(V)$ is dense in $H$ in the sense of Dense Subset of a Topological Space and Real Hilbert Space §topology.

Step 6 (claim 1, completeness). For $m\in\mathbb{N}$, $1/m$ is positive (claim 7 of Elementary Order Arithmetic in an Ordered Field, as $m\ge1>0$), and $1/m=m^{-1}\to0$ by claims 3(a) (with $a=1$) and 1 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities. Let $(h_{m})_{m\in\mathbb{N}}$ be a Cauchy sequence in $(H,d)$ (Cauchy Sequence in a Metric Space). For each $m$ the set $A_{m}=\{v\in V:\ |Jv-h_{m}|<1/m\}$ is nonempty by Step 4 (write $h_{m}=[u]$ and take $v=u_{k}$ for $k$ large). By Axiom of Countable Choice, applied to the family $(A_{m})_{m\in\mathbb{N}}$ of nonempty subsets of $V$, there is a sequence $(v_{m})$ in $V$ with $v_{m}\in A_{m}$ for every $m$. For $m,l\in\mathbb{N}$, Step 3 and the triangle inequality of the metric $d$ (The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric) give

Given $\varepsilon>0$, choose $N_{1}$ with $1/m<\varepsilon/4$ for $m\ge N_{1}$ and $N_{2}$ with $|h_{m}-h_{l}|<\varepsilon/2$ for $m,l\ge N_{2}$; for $m,l\ge\max\{N_{1},N_{2}\}$ the right side is $<\varepsilon/4+\varepsilon/2+\varepsilon/4=\varepsilon$ (claim 3 of Elementary Order Arithmetic in an Ordered Field). So $w=(v_{m})\in C_{\beta}$. By Step 4, $Jv_{m}\to[w]$; choose $N_{3}$ with $|Jv_{m}-[w]|<\varepsilon/2$ for $m\ge N_{3}$ and $N_{4}$ with $1/m<\varepsilon/2$ for $m\ge N_{4}$. For $m\ge\max\{N_{3},N_{4}\}$, $d(h_{m},[w])\le|h_{m}-Jv_{m}|+|Jv_{m}-[w]|<\varepsilon$. Hence $h_{m}\to[w]$, $(H,d)$ is complete (Complete Metric Space), and $H$ is a real Hilbert space by Real Hilbert Space §hilbert.

Step 7 (claim 4, uniqueness). Let $F,G:H\to K$ be continuous with $F(Jv)=G(Jv)=Tv$ for $v\in V$, and let $h=[u]\in H$. By Step 4, $Ju_{k}\to h$. Given $\varepsilon>0$, continuity of $F$ at $h$ (Continuous Map Between Metric Spaces) gives $\delta>0$, and there is $N$ with $d(h,Ju_{k})<\delta$ for $k\ge N$; then $d_{K}(F(Ju_{k}),F(h))<\varepsilon$ for $k\ge N$, so $Tu_{k}=F(Ju_{k})\to F(h)$ in $K$. Likewise $Tu_{k}\to G(h)$, and $F(h)=G(h)$ by Uniqueness of Limits in a Metric Space.

Step 8 (claim 4, existence). Put $C'=C+1>0$. Let $u\in C_{\beta}$. For $k,l\in\mathbb{N}$, linearity of $T$ and the hypothesis give $|Tu_{k}-Tu_{l}|_{K}=|T(u_{k}-u_{l})|_{K}\le C\|u_{k}-u_{l}\|\le C'\|u_{k}-u_{l}\|$ (claim 5 of Elementary Arithmetic in an Ordered Field). Given $\varepsilon>0$, choosing $N$ with $\|u_{k}-u_{l}\|<\varepsilon/C'$ for $k,l\ge N$ gives $|Tu_{k}-Tu_{l}|_{K}<\varepsilon$ (claim 10 of Elementary Order Arithmetic in an Ordered Field); so $(Tu_{k})$ is Cauchy in $K$ and, $K$ being complete (Real Hilbert Space §hilbert), converges to a point $\Lambda(u)\in K$, unique by Uniqueness of Limits in a Metric Space. If $[u]=[u']$, then $u-u'\in N_{\beta}$ (Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §cosets), so $C'\|u_{k}-u'_{k}\|\to0$ by Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §null and claim 3 of Arithmetic of Limits of Real Sequences; given $\varepsilon>0$, for $k$ beyond the two relevant indices, the triangle inequality of $d_{K}$ (The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric), linearity of $T$, the hypothesis on $T$ and $\|u'_{k}-u_{k}\|=\|u_{k}-u'_{k}\|$ (Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §cauchy-schwarz with $c=-1$) give $|Tu'_{k}-\Lambda(u)|_{K}\le|T(u'_{k}-u_{k})|_{K}+|Tu_{k}-\Lambda(u)|_{K}\le C'\|u_{k}-u'_{k}\|+|Tu_{k}-\Lambda(u)|_{K}<\varepsilon/2+\varepsilon/2$. So $Tu'_{k}\to\Lambda(u)$ and $\Lambda(u')=\Lambda(u)$. Hence $\widehat{T}([u])=\Lambda(u)=\lim_{k}Tu_{k}$ is a well-defined map $H\to K$.

For $v\in V$ the sequence $(T\bar v_{k})$ is constant with value $Tv$, which converges to $Tv$ as $d_{K}(Tv,Tv)=0$; so $\widehat{T}(Jv)=Tv$. Linearity: $T(u_{k}+v_{k})=Tu_{k}+Tv_{k}$ and $T(cu_{k})=c\,Tu_{k}$, so by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §linear-limits in $K$ and uniqueness of limits, $\widehat{T}([u]+[v])=\widehat{T}([u+v])=\widehat{T}[u]+\widehat{T}[v]$ and $\widehat{T}(c[u])=c\,\widehat{T}[u]$. Bound: for $h=[u]$, $|Tu_{k}|_{K}\to|\widehat{T}h|_{K}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity, $C\|u_{k}\|\to C|h|$ by (N) and claim 3 of Arithmetic of Limits of Real Sequences, and $|Tu_{k}|_{K}\le C\|u_{k}\|$ for every $k$; so $|\widehat{T}h|_{K}\le C|h|$ by claim 1 of Order Properties of Limits of Real Sequences. Continuity: let $h\in H$ and $\varepsilon>0$, and put $\delta=\varepsilon/C'$. If $d(h,g)<\delta$, then by linearity and the bound, $d_{K}(\widehat{T}g,\widehat{T}h)=|\widehat{T}(g-h)|_{K}\le C|g-h|\le C'd(h,g)<\varepsilon$ (here $|g-h|=d(g,h)=d(h,g)$ by Real Inner Product Space §distance and condition 3 of Metric Space). So $\widehat{T}$ is continuous (Continuous Map Between Metric Spaces), and by Step 7 it is the only continuous map with $\widehat{T}(Jv)=Tv$.

Step 9 (claim 4, inner products). Assume $\langle Tu,Tv\rangle_{K}=\beta(u,v)$ for $u,v\in V$, and let $g=[u]$, $h=[v]$. Then $Tu_{k}\to\widehat{T}g$ and $Tv_{k}\to\widehat{T}h$ in $K$ (Step 8), so $\beta(u_{k},v_{k})=\langle Tu_{k},Tv_{k}\rangle_{K}\to\langle\widehat{T}g,\widehat{T}h\rangle_{K}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity, while $\beta(u_{k},v_{k})\to\widehat\beta(u,v)=\langle g,h\rangle_{H}$ by Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §pairing. By claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences, $\langle\widehat{T}g,\widehat{T}h\rangle_{K}=\langle g,h\rangle_{H}$.