Math223 - Cauchy-Schwarz Inequality Example

Cauchy-Schwarz Inequality Example

Suppose that $V$ be the vector space of continuous function on the closed interval $[0,\ln 3]$. For $f,g \in V$, we define

\[ \left\langle f,g\right\rangle = \int_0 ^{\ln3} e^x f(x)g(x)dx \]

Show that for any $f \in V$

\[ \left( \int_0^{\ln 3} e^x f(x)dx \right )^2 \leq 2 \int_0^{\ln 3} e^x [f(x)]^2 dx \]

We know that:

\[ \int_0^{\ln 3} e^x f(x)dx = \left\langle f,1\right\rangle \]

By cauchy-schuwarz

\[ \left\langle f,1\right\rangle ^2 \leq \left\langle f,f\right\rangle \left\langle 1,1\right\rangle \]

Follows that

\[\begin{aligned} \left( \int_0^{\ln 3} e^x f(x)dx \right )^2 & \leq \int_0 ^{\ln3} e^x f^2(x)dx \cdot \int_0 ^{\ln3} e^x \cdot 1\cdot 1dx \\ \left( \int_0^{\ln 3} e^x f(x)dx \right )^2 & \leq \int_0 ^{\ln3} e^x f^2(x)dx \cdot 2 \end{aligned}\]

Identify those function $f$ in $V$ such that the equality holds.

Cauchy schwartz equality hold

<f,g> = <f,f><g,g> only if f and g are multiples aof eachother. in b) g =1 so equality hold for f=c*1 for all c in R