Question

state and prove bessel inequility
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Answer 1

Answer:

In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element {\displaystyle x}x in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828.[1]

Let {\displaystyle H}H be a Hilbert space, and suppose that {\displaystyle e_{1},e_{2},...}e_1, e_2, ... is an orthonormal sequence in {\displaystyle H}H. Then, for any {\displaystyle x}x in {\displaystyle H}H one has

{\displaystyle \sum _{k=1}^{\infty }\left\vert \left\langle x,e_{k}\right\rangle \right\vert ^{2}\leq \left\Vert x\right\Vert ^{2},}{\displaystyle \sum _{k=1}^{\infty }\left\vert \left\langle x,e_{k}\right\rangle \right\vert ^{2}\leq \left\Vert x\right\Vert ^{2},}

where ⟨·,·⟩ denotes the inner product in the Hilbert space {\displaystyle H}H.[2][3][4] If we define the infinite sum

{\displaystyle x'=\sum _{k=1}^{\infty }\left\langle x,e_{k}\right\rangle e_{k},}x' = \sum_{k=1}^{\infty}\left\langle x,e_k\right\rangle e_k,