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In mathematics, a 1-unconditional normed space (also unconditional normed space) is a specific type of normed vector space with a norm that remains invariant under the arbitrary sign changes of the coordinates of any vector. These spaces play a significant role in functional analysis, particularly in the study of Banach spaces.

Definition

A normed space $(X,\|\cdot \|)$ is said to be 1-unconditional if for every vector $x\in X$ and for every choice of signs $\epsilon _{i}\in \{-1,1\}$, the holds:

$\left\|\sum _{i=1}^{Usersandbox}n\epsilon _{i}x_{i}e_{i}\right\|=\left\|\sum _{i=1}^{n}x_{i}e_{i}\right\|$

where $\{e_{i}\}$ is a basis for $X$, and $x_{i}$ are the coordinates of $x$ with respect to this basis. Equivalently, it holds

$\left\|\sum _{i=1}^{n}x_{i}e_{i}\right\|=\left\|\sum _{i=1}^{n}|x_{i}|e_{i}\right\|$

Properties

• Stability Under Coordinate Changes: The norm in a 1-unconditional normed space does not change if we alter the signs of the coordinates of any vector.
• Example in $\mathbb {R} ^{n}$: The $\ell ^{p}$-norms (for $1\leq p\leq \infty$) on $\mathbb {R} ^{n}$ are examples of 1-unconditional norms. Specifically, for $x\in \mathbb {R} ^{n}$, the norm $\|x\|_{p}=\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{1/p}$ is 1-unconditional.
• Banach Lattices: If $X$ is a Banach lattice, it is often equipped with a 1-unconditional norm, which aligns with the lattice operations.

Applications

• Functional Analysis: 1-unconditional norms are used to study the geometric properties of Banach spaces.
• Approximation Theory: These norms are important in the context of best approximation and in the study of basis properties in Banach spaces.
• Signal Processing: They can be applied in signal processing where stability under sign changes is crucial.

Examples

$\ell ^{p}$-spaces

The space $\ell ^{p}$ (for $1\leq p\leq \infty$) is defined as the set of all sequences $x=(x_{i})$ such that $\|x\|_{p}$ is finite. The norm in these spaces is given by:

• For $1\leq p<\infty$:

 $\|x\|_{p}=\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{1/p}$

• For $p=\infty$:

 $\|x\|_{\infty }=\max _{1\leq i\leq n}|x_{i}|$

These norms are 1-unconditional because changing the signs of the coordinates does not affect the magnitude of the norm.

$c_{0}$-space

The space $c_{0}$ consists of all sequences $x=(x_{i})$ that converge to zero, equipped with the norm:

$\|x\|_{\infty }=\max _{i\in \mathbb {N} }|x_{i}|$

This norm is also 1-unconditional.

See also

• Banach space
• Normed vector space
• Banach lattice
• Functional analysis
• Approximation theory

External links

• Banach Space on Wikipedia
• Functional Analysis on Wikipedia