LaTeX
\|x+y\|\le \|x\|+\|y\|Variables
- x, y: vectors
- ||·||: a norm
How to use this formula
States that the direct displacement is no longer than a path through an intermediate point.
Important notes
- The statement is part of the definition of a norm.
Quick example
For Euclidean vectors, it matches the geometric triangle inequality.
Applicability, worked calculation, and verification
Assumptions and domain checks
- The statement is part of the definition of a norm.
- Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold.
Worked example
For Euclidean vectors, it matches the geometric triangle inequality.
Common mistakes
- Before substituting values into Triangle Inequality for Norms, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established.
Continue the workflow
Use Triangle Inequality for Norms in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\|x+y\|\le \|x\|+\|y\|. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Triangle Inequality for Norms used for?
States that the direct displacement is no longer than a path through an intermediate point.
Can I copy this formula as LaTeX?
Yes. Copy \|x+y\|\le \|x\|+\|y\| or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.