A subspace of $\R^n$ is a subset of $\R^n$ such that it is closed under addition and multiplication by scalars. That is, $V \subset \R^n$ is a vector subspace if $\vec x, \vec y \in V$ and $a\in\R$, then $\vec x + \vec y \in V$ and $a\vec x \in V$.
In $\R^2$, this includes
The dot product of two vectors $\vec x \cdot \vec y$ is $\sum_{i=1}^n x_i y_i$.
$$ \frac{\vec x \cdot \vec y}{|\vec x||\vec y|} = \cos \theta. $$
The cross product in $\R^3$ of two vectors $\vec x \times \vec y$ is
$$ \begin{bmatrix} \det \begin{bmatrix}a_2 & b_2\\a_3&b_3\end{bmatrix}\\ -\det\begin{bmatrix}a_1&b_1\\a_3&b_3\end{bmatrix}\\ \det\begin{bmatrix}a_1&b_1\\a_2&b_2\end{bmatrix} \end{bmatrix} $$
The Cauchy-Schwarz Inequality proves that for two vectors, $|\vec x \cdot \vec y| \leq |\vec x||\vec y|$.
It follows that for matrices, $|AB| \leq |A||B|$.
The triangle inequality proves that $|a+b| \leq |a| - |b|$ since,
$$ |a| = |a+b-b|\\ \leq |a+b|+|-b|\\ =|a+b|+|b|. $$
Generalized, this is,
$$ \lvert\sum_{i=0}^{\infty}x_i\rvert\leq \sum_{i=0}^{\infty}|x_i| $$