## Definition of the Cantor set

The **Cantor ternary set** (named **Cantor set** below) \(K\) is a subset of the real segment \(I=[0,1]\). It is built by induction:

- Starting with \(K_0=I\)
- If \(K_n\) is a finite disjoint union of segments \(K_n=\cup_k \left[a_k,b_k\right]\), \[K_{n+1}=\bigcup_k \left(\left[a_k,a_k+\frac{b_k-a_k}{3}\right] \cup \left[a_k+2\frac{b_k-a_k}{3},b_k\right]\right)\]

And finally \(K=\displaystyle \bigcap_{n \in \mathbb{N}} K_n\). *The Cantor set is created by repeatedly deleting the open middle third of a set of line segments starting with the segment \(I\).*

The Cantor set is a **closed** set as it is an intersection of closed sets. Continue reading Cantor set: a null set having the cardinality of the continuum →

Let’s start by recalling some background about **modules**.

Suppose that \(R\) is a ring and \(1_R\) is its multiplicative identity. A left **\(R\)-module \(M\)** consists of an **abelian group** \((M, +)\) and an operation \(R \times M \rightarrow M\) such that for all \(r, s \in R\) and \(x, y \in M\), we have:

- \(r \cdot (x+y)= r \cdot x + r \cdot y\) (\( \cdot\) is left-distributive over \(+\))
- \((r +s) \cdot x= r \cdot x + s \cdot x\) (\( \cdot\) is right-distributive over \(+\))
- \((rs) \cdot x= r \cdot (s \cdot x)\)
- \(1_R \cdot x= x \)

\(+\) is the symbol for addition in both \(R\) and \(M\).

If \(K\) is a field, \(M\) is \(K\)-**vector space**. It is well known that a vector space \(V\) is having a **basis**, i.e. a subset of **linearly independent** vectors that **spans** \(V\).

*Unlike for a vector space, a module doesn’t always have a basis.* Continue reading A module without a basis →

We consider a **normed vector space** \(E\) over the field of the reals \(\mathbb{R}\) and a **convex** subset \(C \subset E\).

We suppose that \(0 \in C\) and that \(C\) is **unbounded**, i.e. there exists points in \(C\) at distance as big as we wish from \(0\).

The following question arises: “*does \(C\) contains ***a ray**?”. It turns out that the answer depends on the **dimension** of the space \(E\). If \(E\) is of **finite dimension**, then \(C\) always contains a ray, while if \(E\) is of **infinite dimension** \(C\) may not contain a ray. Continue reading An unbounded convex not containing a ray →

We consider a topological vector space \(E\) over the field of the reals \(\mathbb{R}\). The **convex hull** of a subset \(X \subset E\) is the smallest **convex** set that contains \(X\).

The convex hull may also be defined as the intersection of all convex sets containing X or as the set of all **convex combinations** of points in X.

The convex hull of \(X\) is written as \(\mbox{Conv}(X)\). Continue reading A compact whose convex hull is not compact →

In this page \(\mathbb{F}\) refers to a **field**. Given any **vector space** \(V\) over \(\mathbb{F}\), the **dual space** \(V^*\) is defined as the set of all **linear functionals** \(f: V \mapsto \mathbb{F}\). The dual space \(V^*\) itself becomes a vector space over \(\mathbb{F}\) when equipped with the following addition and scalar multiplication:

\[\left\{

\begin{array}{lll}(\varphi + \psi)(x) & = & \varphi(x) + \psi(x) \\

(a \varphi)(x) & = & a (\varphi(x)) \end{array} \right. \] for all \(\phi, \psi \in V^*\), \(x \in V\), and \(a \in \mathbb{F}\).

There is a natural homomorphism \(\Phi\) from \(V\) into the **double dual** \(V^{**}\), defined by \((\Phi(v))(\phi) = \phi(v)\) for all \(v \in V\), \(\phi \in V^*\). This map \(\Phi\) is always **injective**. Continue reading A vector space not isomorphic to its double dual →

## Mathematical exceptions to the rules or intuition