Kernel of a linear functional

Question

I have $L^2((0,1))$ space and functional $$ F(f) = \int_0^1 f(x)(1-x) dx. $$ How do I find the kernel?

---

From definition we want to find $f$ such that $\int_0^1 f(x)(1-x) dx = 0$. I suspect we want the function $f$ to satisfy $f(x) = x f(x)$ a. e. Is this correct? How do I proceed from here?

Answer 1

Try integration by parts. Let h'(x) = g(x) and $g'(x)=f(x)$:

$$\int f(x)(1-x)dx = g(x)(1-x) - \int - g(x) dx = g(x)(1-x) + h(x)$$

So if $h(1)-h(0)-g(0) = 0$ then $\int_{0}^{1} f(x)(1-x)dx = 0$

So take a function $G(x)$ so that:

$$G(1)-G(0)-G'(0)=0$$

Then G''(x) is in $ker(F)$.