P-Norms#
Theorem: P-Norm
For all real numbers \(p \ge 1\), the function \(||\cdot||_p: \mathbb{R}^n \to \mathbb{R}\) defined as
\[||\boldsymbol{x}|| = \left( \sum_{k = 1}^n |x_k|^p \right)^{\frac{1}{p}}\]
for all \(\begin{bmatrix} x_1 & \cdots & x_n \end{bmatrix}^{\mathsf{T}} \in \mathbb{R}^n\) is a norm on the vector space \(\mathbb{R}^n\).
Definition: P-Norm
We call \(||\cdot||_p\) the \(p\)-norm or \(l_p\)-norm on \(\mathbb{R}^n\).
Proof
TODO
Definition: