Bessel's Inequality

• categories: Functional analysis, Theorem

Bessel’s Inequality

Bessel’s inequality is a fundamental result in the theory of Hilbert spaces. It provides an upper bound for the sum of squared coefficients in the expansion of a vector with respect to an orthonormal set. This inequality is particularly useful inFourier analysis and in studying convergence properties of orthogonal expansions.

Statement

Let $H$ be a Hilbert space, and let $\{e_i{\}}_{i\in I}$ be an orthonormal set in $H$. For any vector $v\in H$, we have:

$$\sum _{i\in I}|⟨v,e_i⟩{|}^2\le \Vert v{\Vert }^2.$$

The inner products $⟨v,e_i⟩$ are called the Fourier coefficients of $v$ with respect to the orthonormal set $\{e_i\}$.

Intuition

Bessel’s inequality shows that the “energy” (squared norm) of a vector $v$ in $H$ is at least as large as the sum of the squared magnitudes of its projections onto each vector in any orthonormal set. This guarantees that only a limited amount of the vector’s “energy” can be captured by any subset of orthonormal components, unless the subset spans the entire space.

Implications

1. Convergence of Orthogonal Expansions: If $\{e_i\}$ is a complete orthonormal Basis, then Bessel’s inequality becomes an equality, leading to the Parseval’s Identity:

   $$\sum _{i\in I}|⟨v,e_i⟩{|}^2=\Vert v{\Vert }^2.$$

2. Bounding Fourier Series: In the context of Fourier series, Bessel’s inequality ensures that the sum of the squares of Fourier coefficients is bounded by the $L^2$ norm of the function.

Proof Outline

1. Orthogonal Projection: Consider the finite partial sum $S_n={\sum }_{i=1}^n⟨v,e_i⟩e_i$, which is the projection of $v$ onto the span of $\{e_1,e_2,\dots ,e_n\}$.

2. Pythagorean Theorem: Since $v-S_n$ is orthogonal to each $e_i$ for $i\le n$, by the Pythagorean theorem:

   $$\Vert v{\Vert }^2=\Vert S_n{\Vert }^2+\Vert v-S_n{\Vert }^2.$$

3. Take the Limit: As $n\to \infty$, $\Vert S_n{\Vert }^2$ is given by ${\sum }_{i=1}^n|⟨v,e_i⟩{|}^2$, leading to:

   $$\sum _{i\in I}|⟨v,e_i⟩{|}^2\le \Vert v{\Vert }^2.$$