# Unbounded functional

• Mar 28th 2011, 11:12 PM
raed
Unbounded functional
Dear Colleagues,

The space $\displaystyle C^{1}[a,b]$ is the subspace of $\displaystyle C[a,b]$ consists of all continuously differentiable functions. Let $\displaystyle f$ be a functional defined on $\displaystyle C^{1}[a,b]$ given by $\displaystyle f(x)=x^{'}(c),c=(a+b)/2$ where $\displaystyle x\in C^{1}[a,b]$. Prove that $\displaystyle f$ is not bounded.

Regards,

Raed.
• Mar 28th 2011, 11:27 PM
Drexel28
Quote:

Originally Posted by raed
Dear Colleagues,

The space $\displaystyle C^{1}[a,b]$ is the subspace of $\displaystyle C[a,b]$ consists of all continuously differentiable functions. Let $\displaystyle f$ be a functional defined on $\displaystyle C^{1}[a,b]$ given by $\displaystyle f(x)=x^{'}(c),c=(a+b)/2$ where $\displaystyle x\in C^{1}[a,b]$. Prove that $\displaystyle f$ is not bounded.
I think you can do this on your own. Think about it, what if you created your function to be such that for every $\displaystyle \varepsilon>0$ you create a function $\displaystyle f_\varepsilon\in C^1[a,b]$ such that $\displaystyle \displaystyle f'_\varepsilon\left(\frac{a+b}{2}\right)=\frac{1}{ \varepsilon}$