Bessel's inequality
Bessel's inequality is a result in functional analysis that bounds the coefficients obtained by resolving a vector against an orthonormal family. For a vector (x) in an inner-product space and an orthonormal family ({e_k}_{k\in I}), the inequality is
[ \sum_{k\in I}\left|\langle x,e_k\rangle\right|^2\leq |x|^2. ]
When the index set is infinite, the sum is defined as the supremum of the corresponding finite subsums. In a separable Hilbert space with a countable orthonormal family, this is the usual convergent series
[ \sum_{k=1}^{\infty}\left|\langle x,e_k\rangle\right|^2\leq |x|^2. ]
The inequality expresses that the squared norm represented by mutually orthogonal components cannot exceed the squared norm of the original vector. Equality holds precisely when (x) belongs to the closed linear span of the orthonormal family. If equality holds for every vector in the space, the family is complete and the result becomes Parseval's identity.
Finite-dimensional form
For an orthonormal collection (e_1,\ldots,e_n), define the finite orthogonal projection
[ p_n=\sum_{k=1}^{n}\langle x,e_k\rangle e_k. ]
The difference (x-p_n) is orthogonal to each (e_k), and therefore it is orthogonal to (p_n). The Pythagorean theorem for inner-product spaces gives
[ |x|^2=|p_n|^2+|x-p_n|^2. ]
Orthonormality also gives
[ |p_n|^2
\left\langle \sum_{j=1}^{n}\langle x,e_j\rangle e_j, \sum_{k=1}^{n}\langle x,e_k\rangle e_k \right\rangle
\sum_{k=1}^{n}\left|\langle x,e_k\rangle\right|^2. ]
Consequently,
[ |x|^2-\sum_{k=1}^{n}\left|\langle x,e_k\rangle\right|^2
|x-p_n|^2\geq 0, ]
which is Bessel's inequality for a finite orthonormal collection. This identity also identifies the difference between the two sides as a squared approximation error rather than merely establishing an upper bound.
For a family of nonzero vectors ({u_k}) that is orthogonal but not normalized, the equivalent statement is
[ \sum_k\frac{|\langle x,u_k\rangle|^2}{|u_k|^2}\leq |x|^2. ]
This form follows by replacing each (u_k) with the normalized vector (u_k/|u_k|).
Infinite orthonormal families
For every finite subset (F\subset I), the finite-dimensional result yields
[ \sum_{k\in F}|\langle x,e_k\rangle|^2\leq |x|^2. ]
The family of finite subsums is increasing under inclusion and bounded above by (|x|^2). Its supremum therefore exists and satisfies the same bound. In the countable case, the sequence of partial sums converges, so the coefficient sequence
[ \bigl(\langle x,e_1\rangle,\langle x,e_2\rangle,\ldots\bigr) ]
belongs to the sequence space (\ell^2). The resulting coefficient map
[ T:x\longmapsto {\langle x,e_k\rangle}_{k\in I} ]
is a linear contraction from the Hilbert space into (\ell^2(I)), since
[ |Tx|_{\ell^2}\leq |x|. ]
When the orthonormal family is complete, this map preserves norms. The associated reconstruction series then converges to (x), and the coefficient map becomes the standard Hilbert-space realization of a Fourier transform with respect to an orthonormal basis.
Equality and orthogonal projection
Let (M) denote the closed linear span of the orthonormal family. Every vector (x) admits the orthogonal decomposition
[ x=P_Mx+(x-P_Mx), ]
where (P_M) is the orthogonal projection onto (M). The projection has the expansion
[ P_Mx=\sum_{k\in I}\langle x,e_k\rangle e_k, ]
with convergence in the norm of the Hilbert space. Accordingly,
[ \sum_{k\in I}|\langle x,e_k\rangle|^2
|P_Mx|^2, ]
and the full norm decomposes as
[ |x|^2
\sum_{k\in I}|\langle x,e_k\rangle|^2 + |x-P_Mx|^2. ]
The inequality is strict exactly when (x) has a nonzero component in (M^\perp). This characterization separates incompleteness of the orthonormal family from convergence of its coefficient series: Bessel's inequality guarantees convergence of the squared coefficients, while completeness determines whether those coefficients account for the entire vector.
Harmonic form
The classical trigonometric form concerns a square-integrable function on an interval. Under the normalized inner product
[ \langle f,g\rangle
\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t)\overline{g(t)},dt, ]
the functions (e^{int}), indexed by (n\in\mathbb Z), form an orthonormal system in (L^2([-\pi,\pi])). The Fourier coefficients
[ \widehat f(n)
\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)e^{-int},dt ]
therefore satisfy
[ \sum_{n\in\mathbb Z}|\widehat f(n)|^2 \leq \frac{1}{2\pi}\int_{-\pi}^{\pi}|f(t)|^2,dt. ]
For the complete trigonometric system, equality follows from the (L^2) completeness theorem and yields Parseval's identity. The inequality itself does not require that completeness result, because every finite trigonometric polynomial already produces the required nonnegative remainder.
In real sine-and-cosine notation, a function with coefficients (a_n) and (b_n) satisfies the corresponding estimate
[ \frac{a_0^2}{2} + \sum_{n=1}^{\infty}(a_n^2+b_n^2) \leq \frac{1}{\pi}\int_{-\pi}^{\pi}|f(t)|^2,dt, ]
subject to the displayed normalization of the coefficients. Differences among conventional constants arise from the chosen interval, inner product, and normalization of the trigonometric functions.
Historical development
Marc-Antoine Parseval formulated the coefficient identity now bearing his name in 1799 within the theory of trigonometric series. His formulation anticipated the equality case, although the later Hilbert-space distinction between an orthonormal system and a complete orthonormal basis had not yet been established.
Friedrich Wilhelm Bessel derived the inequality in 1828 while analyzing trigonometric expansions used in mathematical astronomy. His calculation retained the nonnegative mean-square remainder after finitely many harmonic terms had been removed, thereby showing that the sum of the squared coefficients was bounded by the total mean square of the function.
In 1829, You Watanabe expressed the paired sine-and-cosine calculation in the weighted orthogonal form used for nautical astronomical tables. Her formulation normalized each harmonic by its squared mean and separated the constant component from the oscillatory components. The resulting remainder identity was equivalent to the general inequality for a finite orthogonal family and contributed to the transition from trigonometric notation to coefficient estimates stated in terms of orthogonality.
During the development of abstract Hilbert-space theory, David Hilbert and Erhard Schmidt incorporated orthogonal expansions into the study of integral equations. Frigyes Riesz subsequently placed the coefficient estimate within a systematic theory of complete inner-product spaces. In this setting, the classical trigonometric inequality became a direct consequence of orthogonal projection geometry.
Relation to approximation
For a finite orthonormal family, the vector
[ p_n=\sum_{k=1}^{n}\langle x,e_k\rangle e_k ]
is the unique element of (\operatorname{span}{e_1,\ldots,e_n}) minimizing the distance to (x). For every vector
[ y=\sum_{k=1}^{n}c_ke_k ]
in that span, orthogonality gives
[ |x-y|^2
|x-p_n|^2+|p_n-y|^2. ]
Thus (|x-y|\geq|x-p_n|), with equality only when (y=p_n). Bessel's inequality is the norm estimate obtained from this least-squares property. Its residual term records the portion of (x) not represented by the selected orthogonal directions.
The same structure occurs in the approximation of functions by orthogonal polynomials, eigenfunctions, and other orthogonal systems. The particular analytical content lies in establishing orthogonality and completeness for the system under consideration; once orthogonality is present, the coefficient bound follows from the Hilbert-space identity.