BESSEL’S INEQUALITY DR. CASS – FUNCTIONAL ANALYSIS

SCOTT BURLINGTON

Theorem 1 (Bessel’s Inequality). If (e_{n})_{n∈N} is an orthonormal sequence in an inner product space H then, ∀x ∈ H,

∑_{n=1}^{∞} |(x, e_{n})|^{2} ≤ ∥x∥^{2}.

Proof. For N ∈ N let y_{N} = ∑_{n=1}^{N}⟨x, e_{n}⟩e_{n}. By the Closest Point Property of inner product spaces,

∥x − y_{N}∥^{2} = ∥x∥^{2} − ∑_{n=1}^{N} ∥⟨x, e_{n}⟩∥^{2}

and hence

∑_{n=1}^{N} ∥⟨x, e_{n}⟩∥^{2} = ∥x∥^{2} − ∥x − y_{N}∥^{2}
≤ ∥x∥^{2}

Now let N → ∞ in the last inequality.

Theorem 2. Let (e_{n})_{n∈N} be an orthonormal sequence in a Hilbert Space, H and let λ_{n} ∈C, n ∈ N. Then ∑_{n=1}^{∞} λ_{n}e_{n} converges in H iff

∑_{n=1}^{∞} |λ_{n}|^{2} < ∞.

Proof. (⇒) Suppose ∑_{n=1}^{∞} λ_{n}e_{n} = x. For m ≥ k in N,

⟨∑_{n=1}^{m} λ_{n}e_{n}, e_{k}⟩ = ∑_{n=1}^{m} λ_{n}⟨e_{n}, ek⟩ = λ_{k}.

On letting m → ∞ and using the continuity of the inner product we obtain

⟨x, e_{k}⟩ = lim_{m→∞} λ_{k} = λ_{k}.

Date: Sometime in Fall 1994.

By Bessel’s Inequality,

∑_{n=1}^{∞} |λ_{n}|^{2} = ∑_{n=1}^{∞} |⟨s, e_{n}⟩|^{2} ≤ ∥x∥^{2} < ∞.

(⇐) Suppose ∑_{n=1}^{∞} |λ_{n}|^{2} < ∞ and write

x_{m} = ∑_{n=1}^{m} λ_{n}e_{n}.

By Pythagoras’ Theorem, for m, p ∈ N,

∥x_{m+p} − x_{m}∥^{2} = ∥ ∑_{n=m+1}^{m+p} λ_{n}e_{n}∥^{2}

= ∑_{n=m+1}^{m+p} |λ_{n}|^{2} ⟶ 0 as m → ∞.

Thus (x_{m}) is a Cauchy sequence in H, and it therefore converges in H.

University of Western Ontario