Logo

AskSia

Plus

How do I show all p-norms on Euclidean space follow an order?
Feb 5, 2024
How do I show all p-norms on Euclidean space follow an order?
Solution by Steps
step 1
To show that all p-norms on Euclidean space follow an order, we need to understand the definition of p-norms. The p-norm of a vector x∈Rn \mathbf{x} \in \mathbb{R}^n is defined as ∥x∥p=(∑i=1n∣xi∣p)1/p \|\mathbf{x}\|_p = \left( \sum_{i=1}^{n} |x_i|^p \right)^{1/p} for 1 \leq p < \infty
step 2
We need to prove that if 0 < p < q , then ∥x∥p≥∥x∥q \|\mathbf{x}\|_p \geq \|\mathbf{x}\|_q for all x∈Rn \mathbf{x} \in \mathbb{R}^n . This is known as the p-norm inequality
step 3
Apply Hölder's inequality, which states that for any two vectors a,b∈Rn \mathbf{a}, \mathbf{b} \in \mathbb{R}^n and for 1/p+1/q=1 1/p + 1/q = 1 , we have ∑i=1n∣aibi∣≤(∑i=1n∣ai∣p)1/p(∑i=1n∣bi∣q)1/q \sum_{i=1}^{n} |a_i b_i| \leq \left( \sum_{i=1}^{n} |a_i|^p \right)^{1/p} \left( \sum_{i=1}^{n} |b_i|^q \right)^{1/q}
step 4
To use Hölder's inequality, set ai=∣xi∣p a_i = |x_i|^p and bi=1 b_i = 1 for all i i , and let q=p/(p−1) q = p/(p-1) . Then, ∑i=1n∣xi∣p=(∑i=1n∣xi∣p)1/p(∑i=1n1q)1/q \sum_{i=1}^{n} |x_i|^p = \left( \sum_{i=1}^{n} |x_i|^p \right)^{1/p} \left( \sum_{i=1}^{n} 1^q \right)^{1/q}
step 5
Simplify the expression to get (∑i=1n∣xi∣p)1/p≥(∑i=1n∣xi∣p)1/q \left( \sum_{i=1}^{n} |x_i|^p \right)^{1/p} \geq \left( \sum_{i=1}^{n} |x_i|^p \right)^{1/q} because n1/q≥1 n^{1/q} \geq 1 and 1/p < 1/q implies 1/p - 1/q > 0
step 6
Raise both sides of the inequality to the power of p p to obtain ∑i=1n∣xi∣p≥(∑i=1n∣xi∣p)p/q \sum_{i=1}^{n} |x_i|^p \geq \left( \sum_{i=1}^{n} |x_i|^p \right)^{p/q}
step 7
Take the p-th root of both sides to get ∥x∥p≥∥x∥q \|\mathbf{x}\|_p \geq \|\mathbf{x}\|_q , which shows that the p-norms follow an order
Answer
To show that all p-norms on Euclidean space follow an order, we use the definition of p-norms and apply Hölder's inequality to prove that ∥x∥p≥∥x∥q \|\mathbf{x}\|_p \geq \|\mathbf{x}\|_q for 0 < p < q .
Key Concept
p-norm inequality and Hölder's inequality
Explanation
The p-norm inequality is demonstrated by applying Hölder's inequality to show that for any vector in Euclidean space, its p-norms follow an order when 0 < p < q , meaning that as p increases, the p-norm does not increase.
© 2023 AskSia.AI all rights reserved