## Quotient Spaces and the Local Diameter 2 Property With particular focus on \ell_\infty/c_0

##### Abstract

The goal of this thesis is to show that the quotient space \ell_\infty/c_0 has the local diameter 2 property. We will start by defining the quotient space X/Y when X is a vector space and Y is a subspace of X. We will see that when X is normed, then X/Y can be given a norm in a natural way, and that this norm is complete provided the norm in X is. In particular, we have that \ell_\infty/c_0 is a complete quotient space.
We will show that the dual of a quotient space X/Y is isometrically isomorphic to the annihilator of Y in X*, and thus it follows that the dual of \ell_\infty/c_0 is isometrically isomorphic to a subspace of (\ell_\infty)*.
We will realize the dual of \ell_\infty as the space ba(2^N) of finitely additive signed measures on 2^N that are of bounded variation. Furthermore, we will show that the dual space action on ell_\infty is given by the integral of functions in \ell_\infty with respect to such measures. Additionally, we will see that the dual space action on \ell_\infty/c_0 is also given by this integral.
Once the dual space action on \ell_\infty/c_0 is established, we can generate slices S(\varphi,\varepsilon) of the unit ball B_{\ell_\infty/c_0} where \varphi is in S_{(\ell_\infty/c_0)*} and \varepsilon> 0. A slice is a set S(\varphi,\varepsilon):=\{[x]\in B_{\ell_\infty/c_0}: \varphi([x])>1-\varepsilon\}. Furthermore, the diameter of a slice is the maximum distance between elements of the slice. By showing that any slice contains elements [x],[y]\in\ell_\infty/c_0 such that x,y\in\ell_\infty have the values 1,-1 respectively on a set A_j \subset N of infinite cardinality, we are able to show that all slices of B_{\ell_\infty/c_0} have diameter 2, i.e., \ell_\infty/c_0 has the local diameter 2 property.