\section{Algebraic Structure}
\begin{definition}[Euclidean space] For any natural number $n$, the $n$-fold Cartesian product of $\mathbb{R}$ with itself is called a \textit{Euclidean space} and denoted by the symbol $\mathbb{R}^n$.
\[
\mathbb{R}^n\;=\;\{(x_1,x_2,\ldots,x_n) : x_i\in\mathbb{R},\quad 1\leq i\leq n\}
\]
\end{definition}
\begin{definition}[vector sum in Euclidean space] For any $x,y\in\mathbb{R}^n$, define
\[
+:{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^n\quad\mbox{by}\quad x+y\;=\;(x_1+y_1,x_2+y_2,\ldots,x_n+y_n)
\]
\end{definition}
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\begin{definition}[scalar product in Euclidean space] For any $x\in\mathbb{R}^n$ and $\alpha\in\mathbb{R}$, define
\[
:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}^n\quad\mbox{by}\quad \alpha x\;=\;(\alpha x_1,\alpha x_2,\ldots,\alpha x_n)
\]
\end{definition}
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\begin{definition}[inner product in Euclidean space] For any $x,y\in\mathbb{R}^n$, define
\[
\cdot:{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}\quad\mbox{by}\quad x\cdot y\;=\;(x_1y_1+x_2y_2+\cdots+x_ny_n)
\]
\end{definition}
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\begin{definition}[cross product in $\mathbb{R}^3$] For any $x,y\in\mathbb{R}^3$, define
\[
\times:{R}^3\times\mathbb{R}^3\rightarrow\mathbb{R}^3\quad\mbox{by}\quad x\times y\;=\;(x_2y_3-x_3y_2,x_3y_1-x_1y_3,x_1y_2-x_2y_1)
\]
\end{definition}
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\begin{definition}[norms in Euclidean space] For any $x\in\mathbb{R}^n$, define
\[
\|x\|:{R}^n\rightarrow\mathbb{R}\quad\mbox{by}\quad \|x\|\;=\;\sqrt{\sum_{i=1}^n|x_i|^2}
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\]
\[
\|x\|_1:{R}^n\rightarrow\mathbb{R}\quad\mbox{by}\quad \|x\|_1\;=\;\sum_{i=1}^n|x_i|
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\]
\[
\|x\|_\infty:{R}^n\rightarrow\mathbb{R}\quad\mbox{by}\quad \|x\|_\infty;=\;\max\{|x_1|,|x_2|,\ldots,|x_n|\}
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\]
\end{definition}
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\begin{definition}[Euclidean distance] For any $x,y\in\mathbb{R}^n$, define
\[
d:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}\quad\mbox{by}\quad d(x,y)\;=\;\|x-y\|
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\]
\end{definition}
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\begin{theorem} $\mathbb{R}^n$ is a vector space.
\end{theorem}
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\begin{theorem} $"\cdot"$ is an inner product.
\end{theorem}
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\begin{theorem} $\|\cdot\|$ is a norm.
\end{theorem}
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\begin{theorem} $\|\cdot\|_1$ is a norm.
\end{theorem}
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\begin{theorem} $\|\cdot\|_\infty$ is a norm.
\end{theorem}
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\begin{theorem} $d(\cdot,\cdot)$ is a metric.
\end{theorem}
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\begin{theorem}[Cauchy-Schwarz inequality] For any $x,y\in\mathbb{R}^n$,
\[
|x\cdot y| \leq \|x\|\,\|y\|
\]
\end{theorem}
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\begin{theorem} For any $x\in\mathbb{R}^n$,
\[
\|x\|_\infty \leq \|x\| \leq \sqrt{n}\|x\|_\infty
\]
\end{theorem}
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\begin{theorem} For any $x\in\mathbb{R}^n$,
\[
\|x\|\leq \|x\|_1 \leq \sqrt{n}\|x\|
\]
\end{theorem}

