There are four main components to a vector space:

1. A set $V$
2. A field $\mathbb{F}$
3. An operation “$+$” that allows us to add together elements of $V$
4. An operation “$\cdot$” that allows us to multiply, or scale, an element of $V$

Notation ($V\times W$)

Given two sets $V$ and $W$, we can make a new set $V\times W$, which we would call the product of $V$ and $W$, that is: $$\begin{align*} V\times W:=\{(v,w)|v\in V,w\in W\} \end{align*}$$ or that $V\times W$ produces a new set of pairs of elements where the first element is all possible values of $V$ and the same for the second and $W$.

Notation (Set of Functions)

Let $S$ be any set. We use the notation $\mathbb{F}^S$ to denote the set of function from $S$ to $\mathbb{F}$.

Example: Sequences of real numbers: $(a_k)_{k=0}^\infty$, this describes a function $f:\mathbb{N}\mapsto\mathbb{R}$ where $f(k)=a_k$. So $(a_k)_{k=0}^\infty$ can be considered an element of $\mathbb{R}^\mathbb{N}$.

Notation (Polynomials)

A polynomial of one variable, with coefficients in $\mathbb{F}$, is an expression of the form $$\begin{align*} p(t)=a_nt^n+a_{n-1}t^{n-1}+...+a_1t+a_o \end{align*}$$ where $a_i\in\mathbb{F}$, $\forall i=0,1,...n$. The degree of $p$, denoted $deg(p$), is given by $$\begin{align*} deg(p)=max\{k|a_k\not=0\} \end{align*}$$ We use $\mathbb{F}[t]$ to denote the set of all polynomial with coefficients in $\mathbb{F}$, and we use $\mathbb{F}[t]_{\leq n}$ to denoted the set of all polynomials with degree $\leq n$.