The set of all sequences is a vector space.

**Theorem:** The set of all infinite sequences \((a_1, a_2, a_3, \ldots)\) with entries in a field \(\mathbb{F}\) is a vector space over \(\mathbb{F}\).

- **Addition:** \((a_1, a_2, \ldots) + (b_1, b_2, \ldots) = (a_1+b_1, a_2+b_2, \ldots)\)
- **Scalar multiplication:** \(c(a_1, a_2, \ldots) = (ca_1, ca_2, \ldots)\)

This space is often denoted \(\mathbb{F}^\mathbb{N}\) or \(\mathbb{F}^\infty\).

**Subspaces of interest:**
- Bounded sequences (\(\ell^\infty\))
- Convergent sequences
- Sequences with finitely many nonzero terms
- Square-summable sequences (\(\ell^2\))

This space is infinite-dimensional. The standard basis vectors \(\mathbf{e}_i = (0,\ldots,0,1,0,\ldots)\) span only the subspace of sequences with finitely many nonzero terms.Copy