In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers.
The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication.
These are special cases of Lp spaces for the counting measure on the set of natural numbers.
denote the field either of real or complex numbers.
is Fréchet, meaning that it is a complete, metrizable, locally convex topological vector space (TVS).
does not admit a strictly coarser Hausdorff, locally convex topology.
[1] For that reason, the study of sequences begins by finding a strict linear subspace of interest, and endowing it with a topology different from the subspace topology.
The canonical norm induced by this inner product is the usual
is defined to be the space of all bounded sequences endowed with the norm
that when endowed with the supremum norm becomes a Banach space that is denoted by
consisting of all sequences which have only finitely many nonzero elements.
This is not a closed subspace and therefore is not a Banach space with respect to the infinity norm.
becomes a complete, Hausdorff, locally convex, sequential, topological vector space that is not Fréchet–Urysohn.
is also strictly finer than the subspace topology induced on
where every inclusion adds trailing zeros: This shows
Each ℓp is distinct, in that ℓp is a strict subset of ℓs whenever p < s; furthermore, ℓp is not linearly isomorphic to ℓs when p ≠ s. In fact, by Pitt's theorem (Pitt 1936), every bounded linear operator from ℓs to ℓp is compact when p < s. No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of ℓs, and is thus said to be strictly singular.
If 1 < p < ∞, then the (continuous) dual space of ℓp is isometrically isomorphic to ℓq, where q is the Hölder conjugate of p: 1/p + 1/q = 1.
The specific isomorphism associates to an element x of ℓq the functional
Hölder's inequality implies that Lx is a bounded linear functional on ℓp, and in fact
so that the operator norm satisfies In fact, taking y to be the element of ℓp with gives Lx(y) = ||x||q, so that in fact Conversely, given a bounded linear functional L on ℓp, the sequence defined by xn = L(en) lies in ℓq.
The map obtained by composing κp with the inverse of its transpose coincides with the canonical injection of ℓq into its double dual.
For the case of natural numbers index set, the ℓp and c0 are separable, with the sole exception of ℓ∞.
However, since the weak topology on infinite-dimensional spaces is strictly weaker than the strong topology, there are nets in ℓ1 that are weak convergent but not strong convergent.
The question of whether every infinite-dimensional Banach space contains an isomorph of some ℓp or of c0, was answered negatively by B. S. Tsirelson's construction of Tsirelson space in 1974.
The dual statement, that every separable Banach space is linearly isometric to a quotient space of ℓ1, was answered in the affirmative by Banach & Mazur (1933).
That is, for every separable Banach space X, there exists a quotient map
In general, ker Q is not complemented in ℓ1, that is, there does not exist a subspace Y of ℓ1 such that
In fact, ℓ1 has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take
Except for the trivial finite-dimensional case, an unusual feature of ℓp is that it is not polynomially reflexive.
[3] If K is a subset of this space, then the following are equivalent:[3] Here K being equismall at infinity means that for every