LaTeX
|\langle x,y\rangle|^2\le \langle x,x\rangle\langle y,y\rangleVariables
- x, y: vectors in an inner-product space
- ⟨x,y⟩: inner product
How to use this formula
Bounds the absolute inner product by the product of vector norms.
Important notes
- Equality holds exactly when the vectors are linearly dependent.
Quick example
For x=(1,2) and y=(3,4), 11² ≤ 5·25.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Equality holds exactly when the vectors are linearly dependent.
- Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold.
Worked example
For x=(1,2) and y=(3,4), 11² ≤ 5·25.
Common mistakes
- Before substituting values into Cauchy–Schwarz Inequality, 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 Cauchy–Schwarz Inequality 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
|\langle x,y\rangle|^2\le \langle x,x\rangle\langle y,y\rangle. - 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 Cauchy–Schwarz Inequality used for?
Bounds the absolute inner product by the product of vector norms.
Can I copy this formula as LaTeX?
Yes. Copy |\langle x,y\rangle|^2\le \langle x,x\rangle\langle y,y\rangle 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.