Hilbert spaces
General · Mathematics
Study notes
Q: Show ℓ² is a Hilbert space. What is ⟨(1,1/2,...),(1,1/3,...)⟩ in ℓ²? ℓ²: inner product ⟨x,y⟩ = Σxₙȳₙ: complete: HILBERT! ⟨(1,1/2,1/4,...),(1,1/3,1/9,...)⟩ = Σ(1/2ⁿ)(1/3ⁿ) = Σ(1/6ⁿ). = 1/(1-1/6) = 6/5! (Geometric series!) Parallelogram law ✓!