Vectors, Norms, and Inner Products

A vector in $\mathbb{R}^n$ is an ordered tuple of real numbers. Norms measure length and inner products measure alignment.

The Euclidean ($L_2$) norm is $\|x\|_2 = \sqrt{\sum_i x_i^2}$. Other useful norms include the $L_1$ norm $\sum_i |x_i|$ (used in Lasso) and the $L_\infty$ norm $\max_i |x_i|$ (max absolute deviation).

The inner product $\langle x, y \rangle = \sum_i x_i y_i$ measures alignment. Cosine similarity $\langle x, y\rangle / (\|x\| \|y\|)$ ignores magnitude and just captures direction. The Cauchy-Schwarz inequality bounds $|\langle x, y\rangle| \le \|x\|\, \|y\|$, with equality iff the vectors are colinear.

In finance, vectors of asset weights, returns, and factor loadings live in $\mathbb{R}^n$. Norms quantify portfolio risk and turnover; inner products underlie portfolio variance and correlation calculations.