Sharp constants for finite dimensional norms

A general result about deriving sharp constants for finite dimensional \(\ell_{p}\)-norms

linear algebra
math
norms
Author

Shamindra Shrotriya

Published

May 12, 2022

TL;DR

I walk through a cool and possibly less known result for sharp bounds on \(\ell_{p}\) norms in finite dimensional vector spaces. This result enables working with \(\ell_{p}\)-norms directly rather than approximating them with with more widely used \(\ell_{1}, \ell_{2}\), and \(\ell_{\infty}\)-norms, for example. This exposition is based on (Chapter 2, Wendland 2018) 1.

\(\ell_{p}\)-norm on \(\mathbb{R}^{d}\)

Throughout this presentation we will work in the finite dimensional Euclidean space \(\mathbb{R}^{d}\), for some fixed \(d \in \mathbb{N}\). Moreover we will we will use the \(\ell_{p}\)-norm on this space, which is defined as follows:

Definition 1 (\(\ell_{p}\)-norm on \(\mathbb{R}^{d}\)) For a fixed \(p \in [1, \infty]\), the \(\ell_{p}\)-norm as denoted by \(\lVert \cdot \rVert_{\ell_{p}}: \mathbb{R}^{d} \to \mathbb{R}_{\geq 0}\), is defined as follows:

\[ \lVert \mathbf{x} \rVert_{\ell_{p}} := \left(\sum_{i = 1}^{d} |x_{i} |^{p}\right)^{\frac{1}{p}}, \; \text{for each $\mathbf{x} \in \mathbb{R}^{d}$} \tag{1}\]

We note (without proof) that Definition 1 does indeed define a valid norm on \(\mathbb{R}^{d}\). A previous post showed more of these details. A more detailed proof of this can be found in TODO.

Warm up: Relationships between \(\ell_{1}, \ell_{2}\), and \(\ell_{\infty}\)-norms

Our goal will be to prove bounds for any \(p \in [1, \infty]\). However, we we get our bearings by first shifting focus on the three most commonly used \(\ell_{p}\)-norms on \(\mathbb{R}^{d}\). These are the \(\ell_{1}, \ell_{2}\), and \(\ell_{\infty}\)-norms. The main proposition is as follows:

Proposition 1 (\(\ell_{p}\)-norm on \(\mathbb{R}^{d}\)) For a fixed \(p \in [1, \infty]\), the \(\ell_{p}\)-norm as denoted by \(\lVert \cdot \rVert_{\ell_{p}}: \mathbb{R}^{d} \to \mathbb{R}_{\geq 0}\), is defined as follows:

\[ \lVert \mathbf{x} \rVert_{\ell_{p}} := \left(\sum_{i = 1}^{d} |x_{i} |^{p}\right)^{\frac{1}{p}}, \; \text{for each $\mathbf{x} \in \mathbb{R}^{d}$} \tag{2}\]

References

Wendland, Holger. 2018. Numerical Linear Algebra. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge.

Footnotes

  1. Note: The presentation in this post is intentionally verbose. The goal is to give lots of intuition of the key result and its usefulness, and ensure that the proofs are rigorous. It is written with an empathetic mindset to newcomers, and to myself for future reference.↩︎

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Citation

BibTeX citation:
@online{shrotriya2022,
  author = {Shamindra Shrotriya},
  title = {Sharp Constants for Finite Dimensional Norms},
  date = {2022-05-12},
  url = {https://www.shamindras.com/posts/2022-05-12-shrotriya2022sharplpnormconsts},
  langid = {en}
}
For attribution, please cite this work as:
Shamindra Shrotriya. 2022. “Sharp Constants for Finite Dimensional Norms.” May 12, 2022. https://www.shamindras.com/posts/2022-05-12-shrotriya2022sharplpnormconsts.