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.