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

(i) $d(x,y)\geq 0$ is equality if and only if $x=y$;

(ii) $d(x,y)=d(y,x)$;

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

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

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 $X$.

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

Examples