Relevant definitions used throughout Analysis II.

\definition \textbf{Euclidian Norm} $||x|| := \displaystyle\sqrt{\sum_{i=1}^{n} x_i^2}$\\
\subtext{Used to generalize $|x|$ in many Analysis I definitions}

\lemma \textbf{Properties of} $||x||$
\begin{center}
    $
    \begin{array}{ll}
        (i)     & ||x|| \geq 0 \\
        (ii)    & ||x|| \iff x = 0 \\
        (iii)   & ||\alpha x|| = \alpha \cdot ||x|| \\
        (iv)    & ||x + y|| \leq ||x|| + ||y||\quad \text{(Triangle Inequality)}
    \end{array}
    $
\end{center}
\subtext{$\forall x,y \in \R^n,\quad \alpha \in \R\\$}