Bessel's Inequality for Inner Product Spaces

Bessel's Inequality for Inner Product Spaces

Recall from the The Pythagorean Identity for Inner Product Spaces page that if $H$ is an inner product space and $\{ x_1, x_2, ..., x_n \}$ is an orthonormal subset of $H$ then for all $c_1, c_2, ..., c_n \in \mathbb{R}$ we have that:

(1)
\begin{align} \quad \biggr \| \sum_{k=1}^{n} c_kx_k \biggr \|^2 = \sum_{k=1}^{n} |c_k|^2 \end{align}

We now use the Pythagorean identity to prove the very important Bessel's inequality. We first need the following lemma.

 Lemma 1: Let $H$ be an inner product space. If $\{ e_1, e_2, ..., e_n \}$ is an orthonormal set then for all $h \in H$, $\displaystyle{\sum_{k=1}^{n} \langle e_k, h \rangle^2 \leq \| h \|^2}$.
• Proof: Let $\displaystyle{g = \sum_{k=1}^{\infty}\langle e_k, h \rangle e_k}$. Observe that:
(2)
\begin{align} \quad \| g \|^2 = \langle g, g \rangle = \left \langle \sum_{k=1}^{\infty}\langle e_k, h \rangle e_k, \sum_{k=1}^{\infty}\langle e_k, h \rangle e_k \right \rangle = \sum_{k=1}^{n} \langle e_k, h \rangle^2 \| e_k \|^2 = \sum_{k=1}^{n} \langle e_k, h \rangle^2 \quad (*) \end{align}
• For each $h \in H$ we have that:
(3)
\begin{align} \quad 0 \leq \| h - g \|^2 &= \langle h - g, h - g \rangle \\ &= \| h \|^2 - 2 \langle h, g \rangle + \| g \|^2 \\ &= \| h \|^2 - 2 \left \langle h, \sum_{k=1}^{\infty} \langle e_k, h \rangle e_k \right \rangle + \| g \|^2 \\ &= \| h \|^2 - 2 \sum_{k=1}^{\infty} \langle e_k, h \rangle^2 + \| g \|^2 \\ & \overset{(*)} = \| h \|^2 - 2 \| g \|^2 + \| g \|^2 \\ & = \| h \|^2 - \| g \|^2 \end{align}
• Therefore $\| g \|^2 \leq \| h \|^2$ which implies that $\| g \| \leq \| h \|$. In other words:
(4)
\begin{align} \quad \sum_{k=1}^{n} \langle e_k, h \rangle^2 \leq \| h \|^2 \quad \blacksquare \end{align}
 Theorem 2 (Bessel's Inequality): Let $H$ be an inner product space and let $(e_k)_{k=1}^{\infty}$ be an orthonormal sequence. Then for all $h \in H$ we have that $\displaystyle{\sum_{k=1}^{\infty} \langle e_k, h \rangle^2 \leq \| h \|^2}$.
• Proof: For each $n \in \mathbb{N}$ we have by lemma 1 that:
(5)
\begin{align} \quad \sum_{k=1}^{n} \langle e_k, h \rangle^2 \leq \| h \|^2 \end{align}
• Taking the limit as $n \to \infty$ gives us Bessel's inequality. $\blacksquare$