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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 is expressed as

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

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.

Parseval's Identity is given by

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 $[-\pi,\pi]$.

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

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

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.

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

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.

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.

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

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.