Proof of the Generalized Bessel's Inequality in Hilbert Spaces 📂Hilbert Space

Proof of the Generalized Bessel's Inequality in Hilbert Spaces

Theorem¹

If $\left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}}$ is a regular orthogonal set in the Hilbert space $H$, the following holds.

(a) For all $\left\{ c_{k} \right\}_{k \in \mathbb{N}} \in \ell^{2}$, the infinite series $\sum_{k \in \mathbb{N}} c_{k} \mathbf{v}_{k}$ converges.

(b) For all $\mathbf{v} \in H$,

$$ \sum_{k \in \mathbb{N}} \left| \left\langle \mathbf{v} , \mathbf{v}_{k} \right\rangle \right|^{2} \le \left\| \mathbf{v} \right\|^{2} $$

Explanation

A $\ell^{2}$ space is a function space consisting of a set of complex sequences whose sum of squares converges. The Bessel inequality is important in Fourier analysis and can be generalized to any Hilbert space.

Proof

(a)

Let us take $\left\{ c_{k} \right\}_{k \in \mathbb{N}} \in \ell^{2}$. For all natural numbers $n > m$,

$$ \begin{align*} \left\| \sum_{k=1}^{n} c_{k} \mathbf{v}_{k} - \sum_{k=1}^{m} c_{k} \mathbf{v}_{k} \right\|^{2} =& \left\| \sum_{k=m+1}^{n} c_{k} \mathbf{v}_{k} \right\|^{2} \\ =& \left\langle \sum_{k=m+1}^{n} c_{k} \mathbf{v}_{k} , \sum_{l=m+1}^{n} c_{l} \mathbf{v}_{l} \right\rangle \\ =& \sum_{k = m+1}^{n} \sum_{l = m+1}^{n} c_{k} \overline{c_{l}} \left\langle \mathbf{v}_{k} , \mathbf{v}_{l} \right\rangle \end{align*} $$

Due to the regular orthogonality of $\left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}}$,

$$ \left\| \sum_{k=1}^{n} c_{k} \mathbf{v}_{k} - \sum_{k=1}^{m} c_{k} \mathbf{v}_{k} \right\|^{2} = \sum_{k=m+1}^{n} c_{k} \overline{c_{k}} = \sum_{k=m+1}^{n} \left| c_{k} \right|^{2} $$

Since we have $\left\{ c_{k} \right\}_{k \in \mathbb{N}} \in \ell^{2}$, so $\lim_{m \to \infty} \sum_{k=m+1}^{n} \left| c_{k} \right|^{2} = 0$ is true, and $\left\{ \sum_{k =1}^{n} c_{k} \mathbf{v}_{k} \right\}_{n \in \mathbb{N}}$ is a Cauchy sequence of $H$, hence it converges.

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(b)

Based on (a) above, the following existence is guaranteed, and according to the Pythagorean theorem,

$$ \left\| \sum_{k=1}^{\infty} c_{k} \mathbf{v}_{k} \right\|^{2} = \sum_{k=1}^{\infty} \left| c_{k} \right|^{2} $$

Now let’s define $T : \ell^{2} \to H$ as $T \left\{ c_{k} \right\} _{k \in \mathbb{N}} := \sum_{k \in \mathbb{N}} c_{k} \mathbf{v}_{k}$.

Equivalent Conditions of Bessel Sequences
Given the sequences $\left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}} \subset H$ and $B > 0$ in Hilbert space $H$, the following two propositions are equivalent.
$\left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}}$ is a Bessel sequence with Bessel bound $B$.
The operator defined as follows $T$ is linear, bounded, and meets $\left\| T \right\| \le \sqrt{B}$. $$ T : \ell^{2} \to H \\ T \left\{ c_{k} \right\}_{k \in \mathbb{N}} := \sum_{k \in \mathbb{N}} c_{k} \mathbf{v}_{k} $$

Definition of Bessel Sequences
For a sequence $\left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}} \subset H$ in Hilbert space $H$, if there is an $B > 0$ satisfying the following, $\left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}}$ is called a Bessel sequence and $B$ is called the Bessel bound. $$ \sum_{k=1}^{\infty} \left| \left\langle \mathbf{v} , \mathbf{v}_{k} \right\rangle \right|^{2 } \le B \left\| \mathbf{v} \right\|^{2},\quad \forall \mathbf{v} \in H $$

Thus, since $T$ is linear and bounded meeting $\left\| T \right\| = 1$, $\left\{ \mathbf{v}_{k} \right\}_{k \in \mathbb{N}}$ is a Bessel sequence with Bessel bound $B=1$. According to the definition of Bessel sequences, the following holds.

$$ \sum_{k \in \mathbb{N}} \left| \left\langle \mathbf{v} , \mathbf{v}_{k} \right\rangle \right|^{2} \le \left\| \mathbf{v} \right\|^{2} $$

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