Completion Theorem (Inner Product Space)

== Theorem ==
Let $\GF \in \set {\R, \C}$.
Let $\struct {V, \innerprod \cdot \cdot_V}$ be an inner product space over a subfield $\GF$.
Let $\norm {\, \cdot \,}_V$ be the inner product norm induced by $\innerprod \cdot \cdot_V$.
Let $\struct {H, \norm {\, \cdot \,}_H}$ be a completion of $V$ with respect to $\norm {\, \cdot \,}$ given by Completion Theorem (Normed Vector Space).
Then $\innerprod \cdot \cdot_V$ can be extended to an inner product $\innerprod \cdot \cdot_H$ on $H$.
By definition, $H$ will be a Hilbert space.
Therefore, the theorem can alternatively be stated as:
:any inner product space may be completed to a Hilbert space.
== Proof ==
We first show that $\norm {\, \cdot \,}_H$ is induced by an inner product by using Norm satisfying Parallelogram Law induced by Inner Product.
From Completion Theorem (Normed Vector Space), there exists a linear isometry $\phi : V \to H$ such that $\phi \sqbrk V$ is dense in $H$. 
From Parallelogram Law (Inner Product Space), we have:
:$\norm {u + v}_V^2 + \norm {u - v}_V^2 = 2 \paren {\norm u_V^2 + \norm v_V^2}$
for each $u, v \in V$. 
Since $\phi$ is a linear isometry, we have:
:$\norm {u + v}_V = \norm {\map \phi {u + v} }_H = \norm {\map \phi u + \map \phi v}_H$
:$\norm {u - v}_V = \norm {\map \phi {u - v} }_H = \norm {\map \phi u - \map \phi v}_H$
:$\norm u_V = \norm {\map \phi u}_H$
:$\norm v_V = \norm {\map \phi v}_H$
So we have:
:$\norm {\map \phi u + \map \phi v}_H^2 + \norm {\map \phi u - \map \phi v}_H^2 = 2 \paren {\norm {\map \phi u}_H^2 + \norm {\map \phi v}_H^2}$
for each $u, v \in V$.
In other words, we have:
:$\norm {x + y}_H^2 + \norm {x - y}_H^2 = 2 \paren {\norm x^2_H + \norm y^2_H}$ 
for each $x, y \in \phi \sqbrk V$. 
Now let $x, y \in H$.
Since $\phi \sqbrk V$ is dense in $H$, there exists sequences $\sequence {x_n}_{n \mathop \in \N}$, $\sequence {y_n}_{n \mathop \in \N}$ in $\phi \sqbrk V$ such that:
:$x_n \to x$
and:
:$y_n \to x$ 
in $\struct {H, \norm {\, \cdot \,}_H}$, by Point in Closure of Subset of Metric Space iff Limit of Sequence.
For each $n \in \N$, we have:
:$\norm {x_n + y_n}_H^2 + \norm {x_n - y_n}_H^2 = 2 \paren {\norm {x_n}^2_H + \norm {y_n}^2_H}$
By using Norm is Continuous, we have:
:$\norm {x + y}_H^2 + \norm {x - y}^2_H = 2 \paren {\norm x^2_H + \norm y^2_H}$
by taking $n \to \infty$.
=== Case 1: $\GF = \R$ ===
Define:
:$\ds \innerprod x y_H = \frac {\norm {x + y}^2_H - \norm {x - y}^2_H} 4$
for each $x, y \in H$.
By Norm satisfying Parallelogram Law induced by Inner Product: Real Case, we have that $\norm {\, \cdot \,}_H$ is induced by $\norm {\, \cdot \,}_H$. 
Now take $x, y \in \phi \sqbrk V$. 
Then there exists $u, v \in V$ with $x = \map \phi u$, $y = \map \phi v$ so that:
Hence $\innerprod \cdot \cdot_H$ extends $\innerprod \cdot \cdot_V$ up to identifying $V$ and $\phi \sqbrk V$.
=== Case 2: $\GF = \C$ ===
Define:
:$\ds \innerprod x y_H = \frac 1 4 \sum_{k \mathop = 0}^3 i^k \norm {x + i^k y}_H^2$
for each $x, y \in H$. 
By Norm satisfying Parallelogram Law induced by Inner Product: Complex Case, we have that $\norm {\, \cdot \,}_H$ is induced by $\norm {\, \cdot \,}_H$.
Now take $x, y \in \phi \sqbrk V$. 
Then there exists $u, v \in V$ with $x = \map \phi u$, $y = \map \phi v$ so that:
Hence $\innerprod \cdot \cdot_H$ extends $\innerprod \cdot \cdot_V$ up to identifying $V$ and $\phi \sqbrk V$.
== Sources ==
* : $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples: Proposition $1.9$
Category:Hilbert Spaces
Category:Inner Product Spaces
Category:Named Theorems