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Bessel's inequality - Wikipedia

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In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element 𝑥 in a Hilbert space with respect to an orthonormal sequence. The inequality is named for F. W. Bessel, who derived a special case of it in 1828.[1] Conceptually, the inequality is a generalization of the Pythagorean theorem to infinite-dimensional spaces. It states that the "energy" of a vector 𝑥 , given by ‖ 𝑥 ‖ 2 , is greater than or equal to the sum of the energies of its projections onto a set of perpendicular basis directions. The value | ⟨ 𝑥 , 𝑒 𝑘 ⟩ | 2 represents the energy contribution along a specific direction 𝑒 𝑘 , and the inequality guarantees that the sum of these contributions cannot exceed the total energy of 𝑥 . When the orthonormal sequence forms a complete orthonormal basis, Bessel's inequality becomes an equality known as Parseval's identity. This signifies that the sum of the energies of the projections equals the total energy of

Bessel's inequality - Wikipedia Jump to content From Wikipedia, the free encyclopedia Theorem on orthonormal sequences In mathematics , especially functional analysis , Bessel's inequality is a statement about the coefficients of an element x {\displaystyle x} in a Hilbert space with respect to an orthonormal sequence . The inequality is named for F. W. Bessel , who derived a special case of it in 1828. [ 1 ] Conceptually, the inequality is a generalization of the Pythagorean theorem to infinite-dimensional spaces. It states that the " energy " of a vector x {\displaystyle x} , given by ‖ x ‖

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