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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.

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.

Bessel's Inequality is expressed as

Parseval's Identity is given by

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.

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

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

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.

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

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.

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.

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 $|\langle x, e_1 \rangle|^2 + |\langle x, e_2 \rangle|^2 + |\langle x, e_3 \rangle|^2 \leq \|x\|^2$.

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

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.