Category:Cauchy-Bunyakovsky-Schwarz Inequality

This category contains pages concerning Cauchy-Bunyakovsky-Schwarz Inequality:

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 $w_i, z_i \in \C$.

Continuous Linear Transformation Space with Supremum Operator Norm

Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$, $\struct {Z, \norm {\, \cdot \,}_Z}$ be normed vector spaces.

Let $A : Y \to Z$ and $B : X \to Y$ be continuous linear transformations.

Let $\norm {\, \cdot \,}$ be the supremum operator norm.

Let $\circ$ denote the composition.

Then:

$\norm {A \circ B} \le \norm A \cdot \norm B$

Definite Integrals

Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$.

Then:

$\ds \paren {\int_a^b \map f t \, \map g t \rd t}^2 \le \int_a^b \paren {\map f t}^2 \rd t \int_a^b \paren {\map g t}^2 \rd t$