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.
**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.