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Bessel's Inequality and Parseval's Identity

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Bessel's Inequality and Parseval's Identity

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Orthonormal Set - Definition

An orthonormal set is a set of vectors in an inner product space that are all unit vectors and orthogonal to each other.

Parseval's Identity - Example

For a Fourier series expansion of a function, Parseval's Identity implies that the sum of the squares of the coefficients is equal to the integral of the square of the function over the interval $[-\pi,\pi]$ .

Hilbert Space - Definition

A Hilbert space is a complete inner product space, which means it is a space where every Cauchy sequence of vectors has a limit within the space.

Inner Product Space - Definition

An inner product space is a vector space with an additional structure called an inner product, which allows defining angles and lengths.

Projection of a Vector - Explanation

The projection of vector $x$ onto vector $y$ (assuming $y$ is not the zero vector) is given by the formula

\text{Proj}_{y}(x) = \frac{\langle x, y \rangle}{\|y\|^2}y

Bessel's Inequality - Definition

In any inner product space, the sum of the squares of the projections of any vector onto a set of orthonormal vectors is less than or equal to the square of the norm of the vector.

Bessel's Inequality - Mathematical Expression

Bessel's Inequality is expressed as

\sum_{n=1}^{\infty} |\langle x, e_n \rangle|^2 \leq \|x\|^2

where

\{e_n\}

is an orthonormal sequence in a Hilbert space, and

x

is any element of that space.

Bessel's Inequality - Example

Consider a vector $x$ in $\mathbb{R}^3$ and an orthonormal basis consisting of vectors $e_1, e_2, e_3$ . Bessel's Inequality asserts that $|\langle x, e_1 \rangle|^2 + |\langle x, e_2 \rangle|^2 + |\langle x, e_3 \rangle|^2 \leq \|x\|^2$ .

Complete Space - Definition

In the context of metric spaces, a complete space is one in which every Cauchy sequence converges to a limit that is within the space.

Square-summable sequences - Definition

Square-summable sequences are sequences of numbers whose series of squares is convergent. This property is fundamental in the definition of the $l^2$ space.

Parseval's Identity - Mathematical Expression

Parseval's Identity is given by

\sum_{n=1}^{\infty} |\langle x, e_n \rangle|^2 = \|x\|^2

for any element

x

in a Hilbert space and an orthonormal basis

\{e_n\}

of that Hilbert space.

Fourier Series Expansion - Relevance to Parseval's Identity

In the context of Fourier series, Parseval's Identity equates the sum of the squares of the coefficients of the series to the integral of the square of the function, illustrating the conservation of energy between time and frequency domains.

Cauchy Sequence - Definition

A Cauchy sequence is a sequence in a metric space such that for any small positive distance, there's a point in the sequence beyond which all terms of the sequence are closer to each other than that distance.

Orthogonal Vectors - Definition

Two vectors are orthogonal if their inner product is zero, which geometrically implies that they are at right angles to each other.

Parseval's Identity - Definition

Parseval's Identity states that for any two square-summable sequences, the sum of the product of their elements is equal to the square of the norm of their inner product in $l^2$ space.

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