Bessel's Inequality

Theorem

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 $h \in V$:

$\ds \sum_{n \mathop = 1}^\infty \size {\innerprod h {e_n} }^2 \le \norm h^2$

Corollary 1

Let $E$ be a orthonormal subset of $V$.

Then, for each $h \in V$, the set:

$\set {e \in E : \innerprod h e \ne 0}$

is countable.

Corollary 2

Let $E$ be a orthonormal subset of $V$.

Then, for all $h \in V$:

$\ds \sum_{e \mathop \in E} \size {\innerprod h e}^2 \le \norm h^2$

Proof

Note that for any natural number $n$ we have, applying sesquilinearity of the inner product:

\(\ds \norm {h - \sum_{k \mathop = 1}^n \innerprod h {e_k} e_k}^2\)

\(=\)

\(\ds \innerprod {h - \sum_{k \mathop = 1}^n \innerprod h {e_k} e_k} {h - \sum_{j \mathop = 1}^n \innerprod h {e_j} e_j}\)

Definition of Inner Product Norm

\(\ds \)

\(=\)

\(\ds \innerprod h {h - \sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} - \innerprod {\sum_{k \mathop = 1}^n \innerprod h {e_k} e_k} {h - \sum_{j \mathop = 1}^n \innerprod h {e_j} e_j}\)

Definition of Inner Product

\(\ds \)

\(=\)

\(\ds \innerprod h h - \innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} - \innerprod {\sum_{k \mathop = 1}^n \innerprod h {e_k} e_k} h + \innerprod {\sum_{k \mathop = 1}^n \innerprod h {e_k} e_k} {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j}\)

Definition of Inner Product

\(\ds \)

\(=\)

\(\ds {\norm h}^2 - \innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} - \overline {\innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} } + \norm {\sum_{k \mathop = 1}^n \innerprod h {e_k} e_k}^2\)

conjugate symmetry of inner product, Definition of Inner Product Norm

\(\ds \)

\(=\)

\(\ds {\norm h}^2 - \innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} - \overline {\innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} } + \sum_{k \mathop = 1}^n \norm {\innerprod h {e_k} e_k}^2\)

Pythagoras's Theorem for Inner Product Spaces

\(\ds \)

\(=\)

\(\ds {\norm h}^2 - \innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} - \overline {\innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j} } + \sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2\)

since each $e_k$ has $\norm {e_k} = 1$

We have:

\(\ds \innerprod h {\sum_{j \mathop = 1}^n \innerprod h {e_j} e_j}\)

\(=\)

\(\ds \sum_{j \mathop = 1}^n \innerprod h {\innerprod h {e_j} e_j}\)

sesquilinearity of inner product

\(\ds \)

\(=\)

\(\ds \sum_{j \mathop = 1}^n \overline {\innerprod {\innerprod h {e_j} e_j} h}\)

conjugate symmetry of inner product

\(\ds \)

\(=\)

\(\ds \sum_{j \mathop = 1}^n \overline {\innerprod {e_j} h} \overline {\innerprod h {e_j} }\)

\(\ds \)

\(=\)

\(\ds \sum_{j \mathop = 1}^n \innerprod h {e_j} \overline {\innerprod h {e_j} }\)

conjugate symmetry of inner product

\(\ds \)

\(=\)

\(\ds \sum_{j \mathop = 1}^n \size {\innerprod h {e_j} }^2\)

Product of Complex Number with Conjugate

Therefore:

\(\ds \norm {h - \sum_{k \mathop = 1}^n \innerprod h {e_k} e_k}^2\)

\(=\)

\(\ds {\norm h}^2 - \sum_{j \mathop = 1}^n \size {\innerprod h {e_j} }^2 - \overline {\sum_{j \mathop = 1}^n \size {\innerprod h {e_j} }^2} + \sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2\)

\(\ds \)

\(=\)

\(\ds {\norm h}^2 - 2 \sum_{j \mathop = 1}^n \size {\innerprod h {e_j} }^2 + \sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2\)

since $\size {\innerprod h {e_j} }^2$ is real for each $j$, we have $\ds \sum_{j \mathop = 1}^n \size {\innerprod h {e_j} }^2 \in \R$

\(\ds \)

\(=\)

\(\ds {\norm h}^2 - \sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2\)

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:

$\ds {\norm h}^2 \ge \lim_{n \mathop \to \infty} \sum_{k \mathop = 1}^n \size {\innerprod h {e_k} }^2 = \sum_{k \mathop = 1}^\infty \size {\innerprod h {e_k} }^2$

$\blacksquare$

Source of Name

This entry was named for Friedrich Wilhelm Bessel.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 4.$ Orthonormal Sets of Vectors and Bases: $4.8$ Bessel's Inequality