cauchy's inequality: meaning, definition, pronunciation and examples
Very low (technical mathematics)Highly formal, exclusively academic/technical
Quick answer
What does “cauchy's inequality” mean?
A fundamental inequality in mathematics stating that the sum of the products of corresponding entries of two sequences is less than or equal to the product of their norms.
Audio
Pronunciation
Definition
Meaning and Definition
A fundamental inequality in mathematics stating that the sum of the products of corresponding entries of two sequences is less than or equal to the product of their norms.
Specifically, for real numbers (or vectors), the absolute value of the sum of products of corresponding entries is at most the product of the Euclidean norms of the two sequences. It is also known as the Cauchy–Schwarz inequality in its more general vector space form.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Both refer to the same theorem. Spelling and pronunciation of 'inequality' are consistent.
Connotations
None beyond the mathematical context.
Frequency
Extremely low and identical in both varieties, confined to advanced mathematics.
Grammar
How to Use “cauchy's inequality” in a Sentence
[Subject] + satisfies + Cauchy's inequalityOne + can + apply + Cauchy's inequality + to + [mathematical object]It + follows + from + Cauchy's inequality + that + [conclusion]Vocabulary
Collocations
Usage
Meaning in Context
Business
Not used.
Academic
Core terminology in advanced mathematics, physics, and engineering courses. Used in proofs, problem sets, and research papers.
Everyday
Not used.
Technical
Essential in mathematical analysis, linear algebra, signal processing, and quantum mechanics for establishing bounds and relationships.
Vocabulary
Synonyms of “cauchy's inequality”
Strong
Neutral
Weak
Vocabulary
Antonyms of “cauchy's inequality”
Watch out
Common Mistakes When Using “cauchy's inequality”
- Misspelling as 'Cauchy inequality' (missing possessive 's').
- Confusing it with the 'triangle inequality'.
- Incorrectly applying it to sequences that are not square-summable.
- Using it as a verb, e.g., 'We will Cauchy this inequality.'
FAQ
Frequently Asked Questions
For sequences of real or complex numbers, they are essentially the same. The name 'Cauchy-Schwarz' (or Cauchy–Bunyakovsky–Schwarz) is used for the generalisation to inner product spaces.
It is named after Augustin-Louis Cauchy, who published a simple form for finite sums in 1821. Later generalisations are attributed to Viktor Bunyakovsky and Hermann Schwarz.
Yes, it is frequently used in signal processing, control theory, and statistical estimation to derive error bounds and stability criteria.
Because it establishes a relationship of 'less than or equal to' (≤) between two quantities, rather than strict equality (=).
A fundamental inequality in mathematics stating that the sum of the products of corresponding entries of two sequences is less than or equal to the product of their norms.
Cauchy's inequality is usually highly formal, exclusively academic/technical in register.
Cauchy's inequality: in British English it is pronounced /ˈkəʊʃiːz ɪnɪˈkwɒlɪti/, and in American English it is pronounced /ˈkoʊʃiz ˌɪnɪˈkwɑːləti/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of two arrows (vectors). The projection of one onto the other can't be longer than the arrow itself. Cauchy's inequality formalises this geometric idea: 'The shadow is shorter than the object.'
Conceptual Metaphor
MATHEMATICAL BOUNDS ARE PHYSICAL LIMITS (e.g., you cannot get more out of a combination of things than the total 'size' of the things would allow).
Practice
Quiz
In which field is Cauchy's inequality most commonly used?