Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space}, with \reftext{lem:vector-space-basic-identities-2026a}{zero vector} $0_{V}$ and \reftext{def:inner-product-norm-2026a}{induced norm} $\lVert\cdot\rVert$. Let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $[n]$ be the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} determined by $n$, and let $e:[n]\to V$ be an \reftext{def:orthonormal-family-2026a}{orthonormal family}. Let $c:[n]\to\mathbb{C}$ be a map with values $c_{k}$ in the field $\mathbb{C}$ of \reftext{def:complex-numbers-2026a}{complex numbers}, and let $u\in V$.

Sums of vectors are \reftext{def:finite-sum-vector-space-2026a}{finite sums in $V$} and sums of scalars are \reftext{def:finite-sum-field-2026b}{finite sums in a field}; $|z|$ denotes the \reftext{def:complex-modulus-2026a}{modulus} of a complex number $z$ and $\overline{z}$ its \reftext{def:complex-conjugate-2026a}{conjugate}; and $x-y$ abbreviates $x+(-y)$, where $-y$ is the additive inverse of $y$ in $V$. Then the following hold.

\textbf{1. (Coefficients)} For every $j\in[n]$,

\textbf{2. (Norm of a linear combination)}

the sum on the right being a finite sum of \reftext{def:real-numbers-c54-2026c}{real numbers}.

\textbf{3. (Linear independence)} The family $e$ is \reftext{def:linear-independence-finite-family-2026a}{linearly independent}.

\textbf{4. (Orthogonal decomposition)} Put

Then $u=w+p$, and $\langle e_{j},w\rangle=0$ for every $j\in[n]$, and $p$ and $w$ are \reftext{def:orthogonal-vectors-2026a}{orthogonal}, and

\textbf{5. (Bessel's inequality)}

an inequality between real numbers in the order of the \reftext{def:ordered-field-c54-2026b}{ordered field} $\mathbb{R}$.