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

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 spac…