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There are several well defined abstract spaces contrast the familiar Euclidean space, namely Metric spaces, Normed spaces, Function spaces, Hilbert spaces and Banach spaces,  etc. To give a overview of these abstract spaces and potential connections between these spaces. This post is dedicated to that purpose. The post is based on Terrence Tao‘s article ‘Function Spaces’ , Timothy Gowers‘s ‘Normed space and banach spaces’ and ‘Metric spaces’, which are published in Princeton Companion to the Mathematics (PCM). About PCM, I have a short post for it or even much better link for further information. Read the rest of this entry »