ℓp Space

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. Equivalently, it is a function space whose elements are functions from the natural numbers to the field ''K'' 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. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

The most important sequence spaces in analysis are the spaces, consisting of the -power summable sequences, with the ''p''-norm. These are special cases of L^''p'' spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted ''c'' and ''c''₀, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space.

Definition

A sequence $x_ = \left(x_n\right)_$ in a set $X$ is just an $X$-valued map $x_ : \N \to X$ whose value at $n \in \N$ is denoted by $x_n$ instead of the usual parentheses notation $x(n).$

Space of all sequences

Let $\mathbb$ denote the field either of real or complex numbers. The set $\mathbb^$ of all sequences of elements of $\mathbb$ is a vector space for componentwise addition
:$\left(x_n\right)_ + \left(y_n\right)_ = \left(x_n + y_n\right)_,$
and componentwise scalar multiplication
:$\alpha\left(x_n\right)_ = \left(\alpha x_n\right)_.$

A sequence space is any linear subspace of $\mathbb^.$

As a topological space, $\mathbb^$ is naturally endowed with the product topology. Under this topology, $\mathbb^$ is Fréchet, meaning that it is a complete, metrizable, locally convex topological vector space (TVS). However, this topology is rather pathological: there are no continuous norms on $\mathbb^$ (and thus the product topology cannot be defined by any norm). Among Fréchet spaces, $\mathbb^$ is minimal in having no continuous norms:

But the product topology is also unavoidable: $\mathbb^$ does not admit a strictly coarser Hausdorff, locally convex topology. 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.

spaces

For $0 < p < \infty,$ $\ell^p$ is the subspace of $\mathbb^$ consisting of all sequences $x_ = \left(x_n\right)_$ satisfying
$\sum_n , x_n, ^p < \infty.$

If $p \geq 1,$ then the real-valued function $\, \cdot\, _p$ on $\ell^p$ defined by
$\, x\, _p ~=~ \left(\sum_n, x_n, ^p\right)^ \qquad \text x \in \ell^p$
defines a norm on $\ell^p.$ In fact, $\ell^p$ is a complete metric space with respect to this norm, and therefore is a Banach space.

If $p = 2$ then $\ell^2$ is also a Hilbert space when endowed with its canonical inner product, called the , defined for all $x_\bull, y_\bull \in \ell^p$ by
$\langle x_\bull, y_\bull \rangle ~=~ \sum_n \overline y_n.$
The canonical norm induced by this inner product is the usual $\ell^2$-norm, meaning that $\, \mathbf\, _2 = \sqrt$ for all $\mathbf \in \ell^p.$

If $p = \infty,$ then $\ell^$ is defined to be the space of all bounded sequences endowed with the norm
$\, x\, _\infty ~=~ \sup_n , x_n, ,$
$\ell^$ is also a Banach space.

If $0 < p < 1,$ then $\ell^p$ does not carry a norm, but rather a metric defined by
$d(x,y) ~=~ \sum_n \left, x_n - y_n\^p.\,$

''c'', ''c''₀ and ''c''₀₀

A is any sequence $x_ \in \mathbb^$ such that $\lim_ x_n$ exists.
The set of all convergent sequences is a vector subspace of $\mathbb^$ called the . Since every convergent sequence is bounded, $c$ is a linear subspace of $\ell^.$ Moreover, this sequence space is a closed subspace of $\ell^$ with respect to the supremum norm, and so it is a Banach space with respect to this norm.

A sequence that converges to $0$ is called a and is said to . The set of all sequences that converge to $0$ is a closed vector subspace of $c$ that when endowed with the supremum norm becomes a Banach space that is denoted by and is called the or the .

The , is the subspace of $c_0$ 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. For example, the sequence $\left(x_\right)_$ where $x_ = 1/k$ for the first $n$ entries (for $k = 1, \ldots, n$) and is zero everywhere else (that is, $\left(x_\right)_ = \left(1, 1/2, \ldots, 1/(n-1), 1/n, 0, 0, \ldots\right)$) is a Cauchy sequence but it does not converge to a sequence in $c_.$

Space of all finite sequences

Let
:$\mathbb^=\left\$,

denote the space of finite sequences over $\mathbb$. As a vector space, $\mathbb^$ is equal to $c_$, but $\mathbb^$ has a different topology.

For every natural number let $\mathbb^n$ denote the usual Euclidean space endowed with the Euclidean topology and let $\operatorname_ : \mathbb^n \to \mathbb^$ denote the canonical inclusion
:$\operatorname_\left(x_1, \ldots, x_n\right) = \left(x_1, \ldots, x_n, 0, 0, \ldots \right)$.
The image of each inclusion is
:$\operatorname \left( \operatorname_ \right) = \left\ = \mathbb^n \times \left\$
and consequently,
:$\mathbb^ = \bigcup_ \operatorname \left( \operatorname_ \right).$

This family of inclusions gives $\mathbb^$ a final topology $\tau^$, defined to be the finest topology on $\mathbb^$ such that all the inclusions are continuous (an example of a coherent topology). With this topology, $\mathbb^$ becomes a complete, Hausdorff, locally convex, sequential, topological vector space that is Fréchet–Urysohn. The topology $\tau^$ is also strictly finer than the subspace topology induced on $\mathbb^$ by $\mathbb^$.

Convergence in $\tau^$ has a natural description: if $v \in \mathbb^$ and $v_$ is a sequence in $\mathbb^$ then $v_ \to v$ in $\tau^$ if and only $v_$ is eventually contained in a single image $\operatorname \left( \operatorname_ \right)$ and $v_ \to v$ under the natural topology of that image.

Often, each image $\operatorname \left( \operatorname_ \right)$ is identified with the corresponding $\mathbb^n$; explicitly, the elements $\left( x_1, \ldots, x_n \right) \in \mathbb^n$ and $\left( x_1, \ldots, x_n, 0, 0, 0, \ldots \right)$ are identified. This is facilitated by the fact that the subspace topology on $\operatorname \left( \operatorname_ \right)$, the quotient topology from the map $\operatorname_$, and the Euclidean topology on $\mathbb^n$ all coincide. With this identification, $\left( \left(\mathbb^, \tau^\right), \left(\operatorname_\right)_\right)$ is the direct limit of the directed system $\left( \left(\mathbb^n\right)_, \left(\operatorname_\right)_,\N \right),$ where every inclusion adds trailing zeros:
:$\operatorname_\left(x_1, \ldots, x_m\right) = \left(x_1, \ldots, x_m, 0, \ldots, 0 \right)$.
This shows $\left(\mathbb^, \tau^\right)$ is an LB-space.

Other sequence spaces

The space of bounded series, denote by bs, is the space of sequences $x$ for which
:$\sup_n \left\vert \sum_^n x_i \right\vert < \infty.$

This space, when equipped with the norm
:$\, x\, _ = \sup_n \left\vert \sum_^n x_i \right\vert,$

is a Banach space isometrically isomorphic to $\ell^,$ via the linear mapping
:$(x_n)_ \mapsto \left(\sum_^n x_i\right)_.$

The subspace ''cs'' consisting of all convergent series is a subspace that goes over to the space ''c'' under this isomorphism.

The space Φ or $c_$ is defined to be the space of all infinite sequences with only a finite number of non-zero terms (sequences with finite support). This set is dense in many sequence spaces.

Properties of ℓ^''p'' spaces and the space ''c''₀

The space ℓ² is the only ℓ^''p'' space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram law

:$\, x+y\, _p^2 + \, x-y\, _p^2= 2\, x\, _p^2 + 2\, y\, _p^2.$

Substituting two distinct unit vectors for ''x'' and ''y'' directly shows that the identity is not true unless ''p'' = 2.

Each is distinct, in that is a strict subset of whenever ''p'' < ''s''; furthermore, is not linearly isomorphic to when . In fact, by Pitt's theorem , every bounded linear operator from to is compact when . No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of , and is thus said to be strictly singular.

If 1 < ''p'' < ∞, then the (continuous) dual space of ℓ^''p'' is isometrically isomorphic to ℓ^''q'', where ''q'' is the Hölder conjugate of ''p'': 1/''p'' + 1/''q'' = 1. The specific isomorphism associates to an element ''x'' of the functional
$L_x(y) = \sum_n x_n y_n$
for ''y'' in . Hölder's inequality implies that ''L''_''x'' is a bounded linear functional on , and in fact
$, L_x(y), \le \, x\, _q\,\, y\, _p$
so that the operator norm satisfies
:$\, L_x\, _ \stackrel\sup_ \frac \le \, x\, _q.$
In fact, taking ''y'' to be the element of with
:$y_n = \begin 0&\text\ x_n=0\\ x_n^, x_n, ^q &\text~ x_n \neq 0 \end$
gives ''L''_''x''(''y'') = , , ''x'', , _''q'', so that in fact
:$\, L_x\, _ = \, x\, _q.$
Conversely, given a bounded linear functional ''L'' on , the sequence defined by lies in ℓ^''q''. Thus the mapping $x\mapsto L_x$ gives an isometry
$\kappa_q : \ell^q \to (\ell^p)^*.$

The map
:$\ell^q\xrightarrow(\ell^p)^*\xrightarrow$
obtained by composing κ_''p'' with the inverse of its transpose coincides with the canonical injection of ℓ^''q'' into its double dual. As a consequence ℓ^''q'' is a reflexive space. By abuse of notation, it is typical to identify ℓ^''q'' with the dual of ℓ^''p'': (ℓ^''p'')^* = ℓ^''q''. Then reflexivity is understood by the sequence of identifications (ℓ^''p'')^** = (ℓ^''q'')^* = ℓ^''p''.

The space ''c''₀ is defined as the space of all sequences converging to zero, with norm identical to , , ''x'', , _∞. It is a closed subspace of ℓ^∞, hence a Banach space. The dual of ''c''₀ is ℓ¹; the dual of ℓ¹ is ℓ^∞. For the case of natural numbers index set, the ℓ^''p'' and ''c''₀ are separable, with the sole exception of ℓ^∞. The dual of ℓ^∞ is the ba space.

The spaces ''c''₀ and ℓ^''p'' (for 1 ≤ ''p'' < ∞) have a canonical unconditional Schauder basis , where ''e''_''i'' is the sequence which is zero but for a 1 in the ''i''^th entry.

The space ℓ¹ has the Schur property: In ℓ¹, any sequence that is weakly convergent is also strongly convergent . 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 ℓ^''p'' spaces can be embedded into many Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some ℓ^''p'' or of ''c''₀, 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 . That is, for every separable Banach space ''X'', there exists a quotient map $Q:\ell^1 \to X$, so that ''X'' is isomorphic to $\ell^1 / \ker Q$. In general, ker ''Q'' is not complemented in ℓ¹, that is, there does not exist a subspace ''Y'' of ℓ¹ such that $\ell^1 = Y \oplus \ker Q$. In fact, ℓ¹ has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take $X=\ell^p$; since there are uncountably many such ''X''s, and since no ℓ^''p'' is isomorphic to any other, there are thus uncountably many ker ''Q''s).

Except for the trivial finite-dimensional case, an unusual feature of ℓ^''p'' is that it is not polynomially reflexive.

ℓ^''p'' spaces are increasing in ''p''

For p\in ,\infty/math>, the spaces $\ell^p$ are increasing in $p$, with the inclusion operator being continuous: for 1\le p, one has $\, x\, _q\le\, x\, _p$. Indeed, the inequality is homogeneous in the $x_i$, so it is sufficient to prove it under the assumption that $\, x\, _p = 1$. In this case, we need only show that $\textstyle\sum , x_i, ^q \le 1$ for $q>p$. But if $\, x\, _p = 1$, then $, x_i, \le 1$ for all $i$, and then $\textstyle\sum , x_i, ^q \le \textstyle\sum , x_i, ^p = 1$.

''ℓ''² is isomorphic to all separable, infinite dimensional Hilbert spaces

Let H be a separable Hilbert space. Every orthogonal set in H is at most countable (i.e. has finite dimension or $\,\aleph_0\,$). The following two items are related:
* If H is infinite dimensional, then it is isomorphic to ''ℓ''²
* If , then H is isomorphic to $\Complex^N$

Properties of ''ℓ''¹ spaces

A sequence of elements in ''ℓ''¹ converges in the space of complex sequences ''ℓ''¹ if and only if it converges weakly in this space.
If ''K'' is a subset of this space, then the following are equivalent:
# ''K'' is compact;
# ''K'' is weakly compact;
# ''K'' is bounded, closed, and equismall at infinity.
Here ''K'' being equismall at infinity means that for every $\varepsilon > 0$, there exists a natural number $n_ \geq 0$ such that $\sum_^ , s_n , < \varepsilon$ for all $s = \left( s_n \right)_^ \in K$.

See also

* L^p space
*Tsirelson space
*beta-dual space
*Orlicz sequence space
*Hilbert space

References

Bibliography

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Functional analysis
Sequences and series