Cauchy-Bunyakovsky-Schwarz Inequality
Theorem
Semi-Inner Product Spaces
Let $\mathbb K$ be a subfield of $\C$.
Let $V$ be a semi-inner product space over $\mathbb K$.
Let $x, y$ be vectors in $V$.
Then:
- $\size {\innerprod x y}^2 \le \innerprod x x \innerprod y y$
Lebesgue $2$-Space
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f, g: X \to \R$ be $\mu$-square integrable functions, that is $f, g \in \map {\LL^2} \mu$, Lebesgue $2$-space.
Then:
- $\ds \int \size {f g} \rd \mu \le \norm f_2^2 \cdot \norm g_2^2$
where $\norm {\, \cdot \,}_2$ is the $2$-norm.
Complex Numbers
- $\ds \paren {\sum \cmod {w_i}^2} \paren {\sum \cmod {z_i}^2} \ge \cmod {\sum w_i z_i}^2$
where all of $w_i, z_i \in \C$.
Definite Integrals
Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$.
Then:
- $\ds \paren {\int_a^b \map f t \, \map g t \rd t}^2 \le \int_a^b \paren {\map f t}^2 \rd t \int_a^b \paren {\map g t}^2 \rd t$
Also known as
The Cauchy-Bunyakovsky-Schwarz Inequality in its various form is also known as:
- the Cauchy-Schwarz-Bunyakovsky inequality
- the Cauchy-Schwarz inequality
- Schwarz's inequality or the Schwarz inequality
- Bunyakovsky's Inequality or Buniakovski's Inequality.
For brevity, it is sometimes referred to by the abbreviations CS inequality or CBS inequality.
Also see
The special case of the Cauchy-Bunyakovsky-Schwarz Inequality in a Euclidean space is called Cauchy's Inequality.
It is usually stated as:
- $\ds \sum {r_i}^2 \sum {s_i}^2 \ge \paren {\sum {r_i s_i} }^2$
where all of $r_i, s_i \in \R$.
Source of Name
This entry was named for Augustin Louis Cauchy, Karl Hermann Amandus Schwarz and Viktor Yakovlevich Bunyakovsky.