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.

**What is Vector Spaces?**

**What is Metric spaces?**

A function defined on pairs of points (x, y) from a set is called a metric if it has properties (i)–(iii) following. In that case, and together form a metric space.

(i) is equality if and only if ;

(ii) ;

(iii) $latex d(x,y)+d(y,z)\geq d(x,z)$.

The function is translation invariance, which means If x and y are two points and we translate them by adding to both, then their distance should not change: that is, .The second is that the metric should scale correctly, that is for a constant , such that .

**What is the Normed spaces & Banach spaces?**

**What is function spaces?**

A function space have functions with fixed domain and range as its elements (continuous function on [-1,1] etc,) and it is a normed space .

The norm of function in is a function space’s way to measure how large is. It is common, though not universal, for the norm to be defined by a simple formula and for the space to consist precisely of those functions for which the resulting definition makes sense and is finite. Thus, the mere fact that a function belongs to a function space can already convey some qualitative information about that function. For example, it may imply some regularity, decay, boundedness, or integrability on the function . The actual value of the norm makes this information quantitative. It may tell us how regular is, how much decay it has, by which constant it is bounded, or how large its integral is.

**Examples**

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