First Initiation
When we first started to learn mathematics, our concern is only to solve problems, understand what equation means, comprehend what teacher says during class and etc. At high school we learn more about sets and functions. However, if we are not curious enough, then the philosophical meaning of the word “mathematical space” is not emphasized. When $f: \mathbb{R} \rightarrow \mathbb{Q}$ is written, everybody who has learned high school math can say “the function takes values from $\mathbb{R}$ and maps it to $\mathbb{Q}$. But have people reading this ever thought about the background?
What is $\mathbb{Q}$, why do we need $\mathbb{R}$, why we generalize the $\mathbb{R}$ to have $\mathbb{R}^n$. What is topological space, what is metric and the space which use metrics. What is vector space. We will examine them in depth to reveal their reasons for existence and naive formulations, with the aim of establishing a small background for future studies.
Why I do write this? Well, no one taught me about neither the reasons behind the existences of different spaces nor the big picture of them.
Algebraic Structures
For every section I will first give rigorous definition and then intuition.
Set
$\textbf{\small Definition:}$ A set is a collection of objects called elements
We begin with the fundamental unit of modern mathematics: the set. Mathematical objects such as numbers, points, vectors, and functions are collected into these sets, which may be finite or infinite in size. The set serves as the foundational container, or ‘playground,’ for mathematical inquiry. To do mathematics, we take a raw set and equip it with structure (axioms and operations), thereby specializing it into specific spaces.” We first start from algebraic branch and see how objects combine, independent of distance or limits, then continue to topological roots, where we can define nearness and convergence, after we walk on merger topics like normed vector spaces and inner product spaces and as a final part we will talk about complete spaces.
Groups
$\textbf{\small Definition:}$ A nonempty set $G$ with a binary operation $\oplus$ on $G$ is called a group if the following axioms hold: $$ \begin{align} & a\oplus (b \oplus c) = (a\oplus b) \oplus c \\ & \exists 0_G \in G \mid 0_G \oplus a = a\\ & \forall a \in G, \exists a^{\prime} \mid a \oplus a^{\prime} = 0_G \\ \end{align} $$ We denote the group $G$ as $(G, \oplus)$. If $\forall a,b \in G$, $a \oplus b = b \oplus a$ holds then we say $G$ is abelian group. If a given set satisfies the three axioms above, then it gives us a group. For example, take the set integers $\mathbb{Z}$ equipped with operation $+$. We confidently say that it forms a group and written as $(\mathbb{Z}, +)$. Observe that for any $a,b,c$ in $\mathbb{Z}$ we we can write $a+(b+c) = (a+b)+c$ and $0 + a = a$ also for every number $a$ we have $-a$. I think you get the idea but it is great to re-emphasize. The symbol $\oplus$ stands only for arbitrary operations, not for default addition. Let us expand this definition.
Rings
$\textbf{\small Definition:}$ A nonempty set $R$ equipped with two operations $\oplus$ and $\otimes$ is called ring if the following axioms hold: $$ \begin{align} & (R, \oplus) \text{ is abelian group} \tag{1} \\ & a \otimes (b\otimes c) = (a \otimes b) \otimes c \tag{2} \\ & a \otimes (b \oplus c) = (a \otimes b) \oplus (a\otimes c) \tag{3} \end{align} $$ We denote rings as $(R, \oplus, \otimes)$. If $\forall a,b \in R$, $a\otimes b = b\otimes a$ is satisfied then we say $R$ is a commutative ring. If there exists an element $1_R$ such that $a\otimes 1_R = 1_R\otimes a = a$ then we say $R$ is a ring with identity. It is a prerequisite to have a group $(R, \oplus)$ to form a ring. Most famous ring is again $\mathbb{Z}$ equipped with $+$ and $\cdot$ we write $(\mathbb{Z}, +, \cdot)$, with identities $0$ and $1$ for addition and multiplication respectively.
Fields
$\textbf{\small Definition:}$ A nonempty set $F$ equipped with two operations $\oplus$ and $\otimes$ is called field if the following axioms hold: $$ \begin{align} & (F, \oplus) \text{ is abelian group} \tag{1} \\ & (F-{0_F}, \otimes) \text{ is abelian group} \tag{2} \\ & (a \oplus b) \otimes c = (a \otimes c) \oplus (b \otimes c) \tag{3} \\ & a \otimes (b \oplus c) = (a \otimes b) \oplus (a\otimes c) \tag{4} \end{align} $$ So here we need two abelian groups with two distinct operations, namely $(F, \oplus)$ and $(F-0_F, \otimes)$ should be abelian groups. The reason why we choose $F-0_F$ is because $0_F$ has no multiplicative inverse. Thus, we omit. The most famous field is the real numbers $\mathbb{R}$ it is the standart for calculus, physics and regular measurement. Of course $\mathbb{C}$ is also field. Observe that $\mathbb{Z}$ is not a field, because no member of integers have multiplicative inverse. Now that we defined fields using groups and rings we can now define vector spaces.
Famous Spaces
Vector Spaces
$\textbf{\small Definition:}$ A vector space consists of a set $V$ (elements of $V$ are called vectors), a field $F$ (elements of $F$ are called scalars), and two operations vector addition and scalar multiplication that satisfies following axioms:
Let $u, v, w \in V$ and $a, b \in F$ $$ \begin{align} & (u + v) + w = u + (v + w) \tag{1}\\ & \exists 0_V \in V \mid 0 + u = u \tag{2} \\ & \exists (-u)\in V \mid u+(-u)=0 \tag{3} \\ & (a\cdot b) \cdot u = a \cdot (b\cdot u) \tag{4} \\ & (a + b) \cdot u = a\cdot u + b \cdot u \tag{5} \\ & \exists 1_V \in V \mid 1 \cdot u = u \tag{6} \end{align} $$
It is a set built over a field where you can add and scale elements. It gives a basis for classic linear algebra, we built matrix theory upon this space. Without vector spaces we would not able to create AIs. Remember AI is simply matrix multiplication. So before continuing to the norms and lengths I want to introduce topological spaces. As you might have noticed we cannot talk about contuinity, convergence or connectedness. Well, topological spaces will let us investigate these properties.
Topological Spaces
$\textbf{\small Definition:}$ Let $X$ be a nonempty set. A collection $\tau$ of subsets of $X$ is said to be a topology on $X$ if,
$$ \begin{align} & \emptyset, X \in \tau \tag{1} \\ & \{U_i\}_{i \in I} \subseteq \tau &&\Rightarrow \bigcup U_i \in \tau \tag{2}\\ & U_1, \dots, U_n \in \tau &&\Rightarrow \bigcap U_i \in \tau \tag{3} \end{align} $$
Besides these definitions, topology is fundamentally about how points are “related” or “connected” in a space, without needing to know the specific nature of those points or a way to measure distance between them. Topology is an abstract tool that allows us to define convergence, continuity, and limits using only the open sets, without needing any explicit knowledge about the points themselves or a metric. Have you ever wondered how do we say limits are unique in $\mathbb{R}$ (click here!)? It is because $\mathbb{R}$ is an Hausdorff space, which means limits are unique. For the spaces like $T_0$ and $T_1$ we could have nonunique limits.
Kolmogorov Space $(T_0)$
$\textbf{\small Definition:}$ A topological space $(X, \tau)$ is called a $T_0$ space if for any two distinct points $x, y \in X$, there exists an open set $U\in \tau$ such that either: $$ \begin{aligned} & x\in U, y\notin U, \text{or} \\ & y\in U, x\notin U \end{aligned} $$
What basically we are saying is for given any two points, at least one of them has a neighbourhood (an open set) that does not contain the other. This means that the topology is “fine enough” to distinguish between any two points, at least in one direction.
However, it does not require that both points have neighborhoods excluding the other. Maybe $x$ has an open set excluding $y$, but every neighbourhood of $y$ still contains $x$.
As an example to $T_0$, take $X = \{0,1\}$ with topology $\tau = \{\emptyset, \{1\}, X\}$. The open neighbourhoods of $1$ are which is $\{1\}$ and $X$ and open neighbourhood of $0$ is $X$. Observe that $\{1\}$ is open, contains $1$ but not $0$ so condition for $T_0$ holds perfectly.
Fréchet Space $(T_1)$
$\textbf{\small Definition:}$ A topological space $X$ is $T_1$, if for every pair of distinct points $x, y \in X$ with $x \notin y$ $$ \begin{aligned} & \exists U \mid x\in U, y\notin U, \text{and} \\ & \exists V \mid y\in V, x\notin V \end{aligned} $$
Hausdorff Space $(T_2)$
$\textbf{\small Definition:}$ A topological space $X$ is $T_2$ if for every pair of distinct points $x,y \in X$ with $x\notin y$, there exist open sets $U,V \subseteq X$ such that $$ \begin{aligned} &x\in U, y\in V, \text{and} \\ &U \cap V = \emptyset \end{aligned} $$
Metric Spaces
$\textbf{\small Definition:}$ A metric space is a pair $(X, d)$ where $X$ is a nonempty set and $d$ is a function called metric (or distance function) that maps pairs of elements from $X$ to real numbers. $$ \begin{aligned} d: X \times X \rightarrow \mathbb{R} \end{aligned} $$ For $d$ to be valid metric, it must satisfy the following four axioms for all elements $x,y,z \in \X$ $$ \begin{align} &d(x,y) \geq 0 \tag{1} \\ &d(x,y) = 0 \iff x = y \tag{2} \\ &d(x,y) = d(y,x)\tag{3} \\ &d(x,z) \leq d(x,y) + d(y,z) \tag{4} \end{align} $$ So basically, metric space allows us to tell “how far apart” any two points are with a rule $d(x,y)$. Also another inference that we can do is every metric space is a topological space, but not every topological space comes from a metric.
Normed Vector Spaces
$\textbf{\small Definition:}$ Let $V$ be a vector space over the field $\mathbb{R}$ or $\mathbb{C}$. A norm is a function $\lVert \cdot \rVert : V \rightarrow \mathbb{R}$ satisfying the following properties for all vectors $x, y \in V$ and all scalars $\alpha$: $$ \begin{align} &\lVert x \rVert \geq 0 \tag{1} \\ &\lVert x \rVert = 0 \iff x = 0 \tag{2} \\ &\lVert \alpha x \rVert = \vert \alpha\vert \lVert x \rVert \tag{3} \\ &\lVert x + y \rVert \leq \lVert x \rVert + \lVert y \rVert \end{align} $$ The pair $(V, \lVert \cdot \rVert)$ is called a normed vector space.
A normed vector space is a vector space where every vector has a “length” assigned to it. The norm allows us to talk about distances between vectors, making normed spaces a natural setting for analysis and geometry. As you might noticed normed spaces are metric spaces, with metric defined by $d(x,y)=\lVert x - y \rVert$. But for a regular normed spaces we can not sure that every Cauchy sequence in the space converges to a limit that is also in the space. Therefore, if this special feature is satisfied then we say the normed space is Banach space.
Banach Space
$\textbf{\small Definition:}$ A Banach space is a normed vector space that is complete with respect to the metric induced by its norm. This means that every Cauchy sequence in the space converges to a limit that is also in the space. In other words, let $V$ be a vector space over the field $\mathbb{R}$ or $\mathbb{C}$ with a norm $\lVert \cdot \rVert$. The space $V$ is called a Banach space if every Cauchy sequence $(x_n)$ in $V$ converges to some $x \in V$
But, there is more exciting space called $l^p$ spaces, which generalizes the concept of normed spaces by extending the $p$-norm from finite dimensional vector spaces to spaces of functions.
$L^p$ Spaces
$\textbf{\small Definition:}$ The $L^p$ spaces are spaces of functions where the “size” of a function is a measured using a specific formula, called the $p$-norm. For a function $f$, this norm is defined as the $p$-th root of the integral of $\vert f \vert^p$ over some domain, or for $p=\infty$, as the smallest number that bounds the function almost everywhere.
Intuitively, $L^p$ spaces collect all functions whose $p$-norm is finite. The most important feature is that these spaces are complete: every sequence of functions that gets closer and closer together (in the sense of the $p$-norm) will converge to a function that is also in the same space. This completeness is what makes $L^p$ spaces Banach spaces.
Take, for example, $f(x) = x^2$ in the interval $[0,1]$. This function is in $L^p([0,1])$ for every $p \geq 1$ because the integral of $\vert x^2\vert^p$ over $[0,1]$ is finite.
Inner Product Spaces
$\textbf{\small Definition:}$ Let $V$ be a vector space over the field $\mathbb{R}$ or $\mathbb{C}$. An inner product on $V$ is a function $\langle\cdot,\cdot\rangle : V \times V \rightarrow \mathbb{F}$, where $\mathbb{F}$ is the field, satisfying: $$ \begin{align} &\langle x, y \rangle = \overline{\langle y, x\rangle} \tag{1} \\ &\langle ax + by, z \rangle = a \langle x,z \rangle + b\langle y,z \rangle \tag{2} \\ &\langle x,x\rangle \geq 0 \tag{3} \\ &\langle x,x\rangle = 0 \iff x = 0 \tag{4} \\ \end{align} $$ I know, it looks like normed vector space. However, inner product spaces has a great feature that normed spaces does not have: angle calculation. We can measure angles and lengths between vectors, inner product spaces allows us to compute “dot product” for vectors, which gives way to talk about orthogonality, lengths, and projections. A most famous example is of course $\mathbb{R}^n$ equipped with dot product (the standard inner product), but inner product spaces allow these ideas to be extended to spaces of functions or infinite-dimensional spaces. Every inner product is automatically a normed space with the norm defined by $\lVert x \rVert = \sqrt{\langle x, x\rangle}$, and thus metric space, but not every metric/ normed space is inner product space. However, again, if inner product space is also complete then it has a stronger name.
Hilbert Space
$\textbf{\small Definition:}$ A Hilbert space is a vector space equipped with an inner product, and it is complete with respect to the norm induced by that inner product.
By definition a complete inner product space is a Banach space and Cauchy space (complete metric space).
Remember the definition of $L^p$ space. Substitute $p = 2$. Magically, we have $L^2$ spaces, a special type of Hilbert space. A space that we can do angle and length measurements between functions much like vectors in ordinary Euclidean space.