squaresummable

Squaresummable, in mathematics, refers to a sequence (x_n) of real or complex numbers for which the sum of the squares of its terms converges: ∑_{n=1}^∞ |x_n|^2 < ∞. The set of all such sequences is denoted ℓ^2 (or l^2) and forms a Hilbert space under the norm ||x||_2 = sqrt(∑ |x_n|^2) and the inner product ⟨x,y⟩ = ∑ x_n conj(y_n). This structure provides geometric tools for analysis, including notions of distance, angles, and orthogonality.

Key properties include:

- ℓ^2 is complete, so every Cauchy sequence in the norm converges to a limit within ℓ^2.

- The standard basis e_n, where (e_n)_k = δ_{nk}, is an orthonormal basis; finite sequences (with only finitely many nonzero terms) are dense in ℓ^2.

- The space is separable, meaning it has a countable dense subset.

- The dual space of ℓ^2 is isomorphic to ℓ^2 itself (via the Riesz representation theorem).

Examples illustrating the definition:

- If a_n = 1/n^p with p > 1/2, then (a_n) ∈ ℓ^2; in particular, a_n = 1/n belongs to ℓ^2 since ∑ 1/n^2 converges.

- The sequence a_n = 1/√n is not in ℓ^2 because ∑ 1/n diverges.

Connections and applications:

- ℓ^2 is closely related to spaces of square-integrable functions; in Fourier analysis, the sequence of Fourier coefficients of a function in L^2 often lies in ℓ^2, with Parseval’s identity linking norms.

- In signal processing and data compression, square-summable sequences model finite-energy signals.