Cauchy-Schwarz inequality
- Cauchy-Schwarz inequality
-
/koh"shee shwawrts", koh shee"shvahrts"/,
Math.
* * *
Any of several related inequalities developed by Augustin-Louis Cauchy and, later, Herman Schwarz (1843–1921).
The inequalities arise from assigning a
real number measurement, or norm, to the
functions,
vectors, or
integrals within a particular space in order to analyze their relationship. For functions
f and
g, whose squares are integrable and thus usable as a norm, (∫fg)
2/n≤/n(∫f
2)(∫g
2). For vectors
a = (a
1, a
2, a
3,..., a
n) and
b = (b
1, b
2, b
3,..., b
n), together with the inner product (see
inner product space) for a norm, (Σ(a
i, b
i))
2 ≤ Σ(a
i)
2Σ(b
i)
2. In addition to
functional analysis, these inequalities have important applications in
statistics and
probability theory.
* * *
Any of several related inequalities developed by Augustin-Louis Cauchy (
Cauchy, Augustin-Louis, Baron) and, later, Herman Schwarz (1843–1921). The inequalities arise from assigning a
real number measurement, or norm, to the
functions,
vectors, or
integrals within a particular space in order to analyze their relationship. For functions
f and
g, whose squares are integrable and thus usable as a norm, (∫
fg)
2 ≤ (∫
f2)(∫
g2). For vectors a = (
a1,
a2,
a3,…,
an) and b = (
b1,
b2,
b3,…,
bn), together with the inner product (
see inner product space) for a norm, (Σ(
ai,
bi))
2 ≤ Σ(
ai)
2Σ(
bi)
2. In addition to functional analysis, these inequalities have important applications in
statistics and
probability theory.
* * *
Universalium.
2010.
Look at other dictionaries:
Cauchy–Schwarz inequality — In mathematics, the Cauchy–Schwarz inequality, also known as the Schwarz inequality, the Cauchy inequality, or the Cauchy–Schwarz–Bunyakovsky inequality, is a useful inequality encountered in many different settings, such as linear algebra… … Wikipedia
Cauchy-Schwarz inequality — /koh shee shwawrts , koh shee shvahrts /, Math. See Schwarz inequality (def. 2) … Useful english dictionary
Cauchy-Schwarz inequality — noun A theorem which states that the absolute value of the dot product between two vectors is less than or equal to the product of the magnitudes of the two vectors … Wiktionary
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Schwarz inequality — Math. 1. Also called Cauchy s inequality. the theorem that the inner product of two vectors is less than or equal to the product of the magnitudes of the vectors. 2. Also called Cauchy Schwarz inequality. the theorem that the square of the… … Useful english dictionary
Schwarz inequality — noun Cauchy Schwarz inequality … Wiktionary
Cauchy's inequality — Math. See Schwarz inequality (def. 1). [named after A. L. CAUCHY] * * * … Universalium
Cauchy's inequality — Math. See Schwarz inequality (def. 1). [named after A. L. CAUCHY] … Useful english dictionary
Cauchy-Schwarz-Abschätzung — Die Cauchy Schwarz Ungleichung, auch bekannt als schwarzsche Ungleichung oder Cauchy Bunjakowski Schwarz Ungleichung, ist eine nützliche Ungleichung, die in vielen Bereichen der Mathematik verwendet wird, z. B. in der Linearen Algebra (Vektoren) … Deutsch Wikipedia
Cauchy-Schwarz-Ungleichung — Die Cauchy Schwarz Ungleichung, auch bekannt als schwarzsche Ungleichung oder Cauchy Bunjakowski Schwarz Ungleichung, ist eine nützliche Ungleichung, die in vielen Bereichen der Mathematik verwendet wird, z. B. in der Linearen Algebra (Vektoren) … Deutsch Wikipedia
