Table of Contents

## Is L Infinity a separable space?

. We say that a space is separable if it has a countable dense subset.

### How do you prove a space is not separable?

Uncountable Discrete Space is not Separable

- Then T is not separable.
- By definition, T is separable if and only if there exists a countable subset of S which is everywhere dense in T.
- Let H⊆S be everywhere dense in T.
- Then by definition of everywhere dense, H−=S where H− denotes the closure of H.

#### Which spaces are separable?

Separable spaces

- Every compact metric space (or metrizable space) is separable.
- Any topological space that is the union of a countable number of separable subspaces is separable.
- The space of all continuous functions from a compact subset to the real line.

**What is the L infinity norm?**

L-Infinity Norm. The largest absolute value of components of a vector, i.e., L-Infinity norm of a vector.

**What is L infinity space in functional analysis?**

It is the space of all essentially bounded functions. The space of bounded continuous functions is not dense in .

## What is the opposite of separable?

Opposite of able to be separated. indivisible. inseparable. annexable. combinable.

### Are all metric spaces separable?

Abstract. We first show that in the function realizability topos RT(K2) every metric space is separable, and every object with decidable equality is countable. More generally, working with synthetic topology, every T0-space is separable and every discrete space is countable.

#### What is L ∞ and how to find it?

More precisely, L ∞ is defined based on an underlying measure space, (S, Σ, μ). Start with the set of all measurable functions from S to R which are essentially bounded, i.e. bounded except on a set of measure zero.

**Is L ∞ a reflexive Banach space?**

is not a reflexive Banach space . L ∞ is a function space. Its elements are the essentially bounded measurable functions. More precisely, L ∞ is defined based on an underlying measure space, (S, Σ, μ). Start with the set of all measurable functions from S to R which are essentially bounded, i.e. bounded except on a set of measure zero.

**Is $x_q$ separable?**

The $x_Q$ are uncountable and any two elements of this collection are distance 1 apart. We have just shown that $\\ell^\\infty$ is not separable.

## What are the applications of ℓ ∞ and L ∞?

One application of ℓ ∞ and L ∞ is in economies with infinitely many commodities. In simple economic models, it is common to assume that there is only a finite number of different commodities, e.g. houses, fruits, cars, etc., so every bundle can be represented by a finite vector, and the consumption set is a vector space with a finite dimension.