Theorem: Bessel's Inequality

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.
Let $\norm \cdot$ be the inner product norm for $\struct {V, \innerprod \cdot \cdot}$.
Let $E = \set {e_n: n \in \N}$ be a countably infinite orthonormal subset of $V$.
Then, for all $h \in V$:
:$\ds \sum_{n \mathop = 1}^\infty \size {\innerprod h {e_n} }^2 \le \norm h^2$

Proof:
Note that for any natural number $n$ we have, applying sesquilinearity of the inner product: 
{{begin-eqn}}
{{eqn | l = \norm {h - \sum_{k \mathop = 1}^n \innerprod h {e_k} e_k}^2
      | r = \innerprod {h - \sum_{k \mathop = 1}^n \innerprod h {e_k} e_k} {h - \sum_{j \mathop = 1}^n \innerprod h {e_j} e_j}
      | c = {{Defof|Inner Product Norm}}
}}
{{eqn | r = \innerprod h {h - \sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} - \innerprod {\sum_{k \mathop = 1}^n \innerprod h {e_k} e_k} {h - \sum_{j \mathop = 1}^n \innerprod h {e_j} e_j}
      | c = {{Defof|Inner Product}}
}}
{{eqn | r = \innerprod h h - \innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} - \innerprod {\sum_{k \mathop = 1}^n \innerprod h {e_k} e_k} h + \innerprod {\sum_{k \mathop = 1}^n \innerprod h {e_k} e_k} {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j}
      | c = {{Defof|Inner Product}}
}}
{{eqn | r = {\norm h}^2 - \innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} - \overline {\innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} } + \norm {\sum_{k \mathop = 1}^n \innerprod h {e_k} e_k}^2
      | c = conjugate symmetry of inner product, {{Defof|Inner Product Norm}}
}}
{{eqn | r = {\norm h}^2 - \innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} - \overline {\innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} } + \sum_{k \mathop = 1}^n \norm {\innerprod h {e_k} e_k}^2
      | c = Pythagoras's Theorem for Inner Product Spaces
}}
{{eqn | r = {\norm h}^2 - \innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} - \overline {\innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} } + \sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2
      | c = since each $e_k$ has $\norm {e_k} = 1$
}}
{{end-eqn}}
We have: 
{{begin-eqn}}
{{eqn | l = \innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j}
      | r = \sum_{j \mathop = 1}^n \innerprod h {\innerprod h {e_j} e_j}
      | c = sesquilinearity of inner product
}}
{{eqn | r = \sum_{j \mathop = 1}^n \overline {\innerprod {\innerprod h {e_j} e_j} h}
      | c = conjugate symmetry of inner product
}}
{{eqn | r = \sum_{j \mathop = 1}^n \overline {\innerprod {e_j} h} \overline {\innerprod h {e_j} }
}}
{{eqn | r = \sum_{j \mathop = 1}^n \innerprod h {e_j} \overline {\innerprod h {e_j} }
      | c = conjugate symmetry of inner product
}}
{{eqn | r = \sum_{j \mathop = 1}^n \size {\innerprod h {e_j} }^2
      | c = Product of Complex Number with Conjugate
}}
{{end-eqn}}
Therefore: 
{{begin-eqn}}
{{eqn | l = \norm {h - \sum_{k \mathop = 1}^n \innerprod h {e_k} e_k}^2
      | r = {\norm h}^2 - \sum_{j \mathop = 1}^n \size {\innerprod h {e_j} }^2 - \overline {\sum_{j \mathop = 1}^n \size {\innerprod h {e_j} }^2} + \sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2
}}
{{eqn | r = {\norm h}^2 - 2 \sum_{j \mathop = 1}^n \size {\innerprod h {e_j} }^2 + \sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2
      | c = since $\size {\innerprod h {e_j} }^2$ is real for each $j$, we have $\ds \sum_{j \mathop = 1}^n \size {\innerprod h {e_j} }^2 \in \R$
}}
{{eqn | r =  {\norm h}^2 - \sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2
}}
{{end-eqn}}
Since:
:$\ds \norm {h - \sum_{k \mathop = 1}^n \innerprod h {e_k} e_k}^2 \ge 0$
we have: 
:$\ds \sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2 \le {\norm h}^2$
Since:
:$\size {\innerprod h {e_k} }^2 \ge 0$ for each $k$
we have that: 
:the sequence $\ds \sequence {\sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2}_{n \in \N}$ is increasing.
So:
:the sequence $\ds \sequence {\sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2}_{n \in \N}$ is bounded and increasing.
So from Monotone Convergence Theorem (Real Analysis): Increasing Sequence, we have that: 
:the sequence $\ds \sequence {\sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2}_{n \in \N}$ converges. 
Since: 
:$\ds \sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2 \le {\norm h}^2$ for each $n$
we then have from Limits Preserve Inequalities: 
:$\ds {\norm h}^2 \ge \lim_{n \mathop \to \infty} \sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2 = \sum_{k \mathop = 1}^\infty \size {\innerprod h {e_k} }^2$
{{qed}}
{{Namedfor|Friedrich Wilhelm Bessel|cat = Bessel}}