Continuous dual of normed spaces
General · Mathematics
Study notes
Q: Find the dual of ℓ¹. What functional does (1,1/2,1/3,...) define? (ℓ¹)* = ℓ∞! a = (1,1/2,...) ∈ ℓ∞: f(x) = Σaₙxₙ. |f(x)| ≤ ‖a‖∞‖x‖₁: bounded, ‖f‖ = ‖a‖∞ = 1! Riesz (Hilbert): every functional = inner product! (Dual: measuring with sequences!)