Mathematical Space Introduction

Posted March 21, 2017 | View revision history

The analysis will let you know when the common mathematical rules(e.g. Interchanging sums/integrals/limits/derivatives) are justified, and when they are illegal.

We start here by introduce the mathematical space, these concept will provide deeper insight for \(L^2\) functions.

Set Notations and Terminology

The words collection, family, and class will be used synonymously with set.

Countable set

A collection of distinguishable objects is countably infinite if the objects can be put into one-to-one correspondence with the positive integers. Stated more intuitively, the collection is countably infinite if the set of elements can be arranged as a sequence a1, a2, . . . ,. A set is countable if it contains either a finite or countably infinite set of elements.
e.g.: The set of rational numbers are countable; the set of real numbers in [0, 1) is uncountable.

Cartesian product

The cartesian product \(A_1 \times\dotsm\times A_n\) of the sets \(A_1,\dotsc ,A_n\) is the set of all ordered n-tuples\((a_1,\dotsc,a_n)\) where \(a_i\in A_i\) for \(i=1,\dotsc,n\).
The real number system is \(R^1\), and \(R^k = R^1 \times\dotsm\times R^1\). The extended real number system is \(R^1\) with \(-\infty\) and \(\infty\).

Supremum and Infimum

If \(E\subset[-\infty,\infty]\) and \(E\ne\varnothing\), the least upper bound(supremum) and greatest lower bound(infimum) of \(E\) exist in \([-\infty,\infty]\) and are denoted by \(\sup E\) and \(\inf E\).

Topological Space

In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance. In modern mathematics spaces are defined as sets with some added structure. For instance, the space of real numbers comes with operations such as addition and multiplication.

Topological space

A topological space is a pair \((X,F)\), where \(X\) is a set, and \(F \subset 2^X\) is a collection of subsets of \(X\), whose elements are referred to as open sets. Furthermore:

The empty set \(\varnothing\) and the whole set \(X\) are open; in other words, \(\varnothing \in F\) and \(X \in F\).
Any finite intersection of open sets is open;in other words, if \(V_i\in F\) for \(i=1,\dotsc,n\), then \(V_1 \cap V_2\cap\dotsm\cap V_n\in F\).
Any arbitrary union of open sets is open(including infinite unions); in other words, if \((V_\alpha)_{\alpha\in I}\) is a family of sets in \(F\)(finite, countable, ro uncountable), then \(\bigcup_{\alpha\in I} V_{\alpha}\).

\(X\) is called a topological space when the collection \(F\) of open sets can be deduced from context. \(F\) is said to be a topology in \(X\).

Open sets

The members of \(F\) are called the open sets in \(X\).

Neighbourhood

Let \((X,F)\) be a topological space, and let \(x\in X\). A neighbourhood of \(x\) is defined to be any open set in \(F\) which contains \(x\).

Topological convergence

Let \(m\) be an integer, \((X,F)\) be a topological space and let \((x^{(n)})_{n=m}^{\infty}\) be a sequence of points in \(X\). Let \(x\) be a point in \(X\). We say that \((x^{(n)})_{n=m}^{\infty}\) converges to \(x\) if and only if, for every neighbourhood \(V\) of \(x\), there exists an \(N\ge m\) such that \(x^{(n)} \in V\) for all \(n\ge N\).

Continuous functions

If \(X\) and \(Y\) are topological spaces and if \(f\) is a mapping of \(X\) into \(Y\), then \(f\) is said to be continuous provided that \(f^{-1}(V)\) is an open set in \(X\) for every open set \(V\) in \(Y\).

Metric spaces

A metric space \((X,d)\) is a space \(X\) of objects(called points), together with a distance function of metric \(\{ d:X\times X\to [0,+\infty) \}\), which associates to each pair \(x,y\) of points in \(X\) a non-negative real number \(d(x,y)\ge 0\). Furthermore:

For any \(x\in X\), we have \(d(x,x)=0\).
(Positivity) For any distinct \(x,y\in X\), we have \(d(x,y)\gt 0\).
(Symmetry) For any \(x,y\in X\), we have \(d(x,y)=d(y,x)\).
(Triangle inequality) For any \(x,y,z\in X\), we have \(d(x,z)\le d(x,y)+d(y,z)\).

Complete metric spaces

A metric space \((X,d)\) is said to be complete iff every Cauchy sequence in \((X,d)\) is in fact convergent in \((X,d)\).

Compact metric spaces

A metric space \((X,d)\) is said to be compact iff every sequence in \((X,d)\) has at least one convergent subsequence. A subset \(Y\) of a metric space \(X\) is said to be compact if the subspace \((Y,d|_{Y \times Y})\) is compact.

Continuous functions

Let \((X,d_x)\) be a metric space, and let \((Y,d_Y)\) be another metric space, and let \(\{f : X \rightarrow Y \}\) be a function. If \(x_0 \in X\), we say that \(f\) is continuous at \(x_0\) iff for every \(\varepsilon \gt 0\), there exists a \(\delta \gt 0\) such that \(d_Y(f(x),f(x_0)) \lt \varepsilon\) whenever \(d_x(x,x_0) \gt \delta\). We say that \(f\) is continuous iff it is continuous at every point \(x \in X\).

Uniform Continuity

Let \(\{ f : X \rightarrow Y \}\) be a map from one metric space \((X,d_X)\) to another \((Y,d_Y)\). We say that \(f\) is uniformly continuous if, for every \(\varepsilon \gt 0\), there exists a \(\delta \gt 0\) such that \(d_Y(f(x),f(x')) \lt \varepsilon\) whenever \(x,x' \in X\) are such that \(d_X(x,x') \lt \delta\).

Euclidean spaces

Let \(n\ge 1\) be a natural number, and let \(R^n\) be the space of \(n\)-tuple of real numbers: \(R^n=\{(x_1,x_2,\dotsm,x_n):x_1,x_2,\dotsm,x_n\in R\}\).

We define the Euclidean metric(also called the \(l^2\) metric) \(d_{l^2}:R^n\times R^n \rightarrow R\) by \(\eqalign{d_{l^2}((x_1,x_2,\dotsm,x_n),(y_1,y_2,\dotsm,y_n)):=\sqrt{(x_1-y_1)^2+\dots +(x_n-y_n)^2}=(\sum_{i=1}^n (x_i-y_i)^2)^{1/2}}\).

Measurable Space

A collection \(F\) of subsets of a set \(X\) is said to be a \(\sigma\text{-field}\)(or \(\sigma\text{-algebra}\)), with the following properties:

\(\varnothing \in F\).
If \(A\in F\),then \(A^c\in F\).
If \(A_i \in F\) for every \(i\in \mathbb{N}\), then \(\bigcup_{i=1}^{\infty} A_i\in F\).

If \(F\) is a \(\sigma\)-field in \(X\), then \(X\) is called a measurable space, and the members of \(F\) are called the measurable sets in \(X\).

Measurable functions

If \(X\) is a measurable space, \(Y\) is a topological space, and \(f\) is a mapping of \(X\) into \(Y\), then \(f\) is said to be measurable provided that \(f^{-1}(V)\) is a measurable set in \(X\) for every open set \(V\) in \(Y\).

Reference:

Robert G.Gallager. (2009). Principles of Digital Communication (New York: Cambridge University Press).
Terence Tao. (2009). Analysis I. Analysis II (Hindustan Book Agency)
Walter Rudin. (1987). Real and complex analysis (McGraw-Hill Book Company)

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