L^p(μ) is a Banach space for all 1 ≤ p ≤ ∞. Equivalently: every absolutely summable series Σ‖f_n‖_p < ∞ converges in L^p. On L²([0,2π]) with the Fourier basis, a sequence (c_n) ∈ ℓ² determines a unique f with coefficients c_n.
L^p(μ) is a Banach space for all 1 ≤ p ≤ ∞. Equivalently: every absolutely summable series Σ‖f_n‖_p < ∞ converges in L^p. On L²([0,2π]) with the Fourier basis, a sequence (c_n) ∈ ℓ² determines a unique f with coefficients c_n.