proof of Riesz representation theorem for separable Hilbert spaces

Let {𝐞0,𝐞1,𝐞2,…} be an orthonormal basisMathworldPlanetmath for the Hilbert spaceMathworldPlanetmath β„‹. Define

ci=f⁒(𝐞i)   and   v=βˆ‘k=0ncΒ―i⁒𝐞i.

The linear map ( f is continuousMathworldPlanetmath if and only if it is bounded, i.e. there exists a constant C such that |f⁒(v)|≀C⁒βˆ₯vβˆ₯. Then


Simplifying, βˆ‘k=0n|ck|2≀C2. Hence βˆ‘k=0∞ck⁒𝐞k converges to an element u in H.

For every basis element, f⁒(𝐞i)=ck=⟨u,𝐞i⟩. By linearity, it will also be true that

f⁒(v)=⟨u,v⟩⁒ if v is a finite superposition of basis vectors.

Any vector in the Hilbert space can be written as the limit of a sequence of finite superpositions of basis vectors hence, by continuity,

f⁒(v)=⟨u,v⟩⁒ for all ⁒vβˆˆβ„‹

It is easy to see that u is unique. Suppose there existed two vectors u1 and u2 such that f⁒(v)=⟨u1,v⟩=⟨u2,v⟩. Then ⟨u1-u2,v⟩=0 for all vectors vβˆˆβ„‹. But then, ⟨u1-u2,u1-u2⟩=0 which is only possible if u1-u2=0, i.e. if u1=u2.

Title proof of Riesz representation theorem for separable Hilbert spaces
Canonical name ProofOfRieszRepresentationTheoremForSeparableHilbertSpaces
Date of creation 2013-03-22 14:34:20
Last modified on 2013-03-22 14:34:20
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 6
Author asteroid (17536)
Entry type Proof
Classification msc 46C99