To solve problem about a bounded linear functional, I am having a problem with the Hahn Banach Theorem.
Problem is:
For $n \in \mathbb{N}$ and $1 \leq p < \infty$, let $X_n \subset L^p([0,1]$ be a collection of polynomials of degree$\leq n$ and let $X = \cup_{n=1}^\infty X_n$.
Is there a bounded linear functional $\Lambda$ on $L^p([0,1]$ such that $\Lambda(f) = f'(0)$ for all $f \in X$?
First I show that for fixed $n \in \mathbb{N}$,
Consider $\Lambda : X_n \rightarrow \mathbb{R}$ which satisfies $\Lambda(f) = f'(0)$. Since $f \in X_n$, we can express $f$ as $f(x) = a_nx^n + \cdots + a_1x + a_0$. $a_i$ are real numbers. By using fact that a derivative as a linear functional$(D)$ is linear and bounded, we know that $\Lambda$ is a bounded linear functional. Therefore we can use the Hahn Banach Theorem, that there exists an extension $\Lambda^* : L^p([0,1]) \rightarrow \mathbb{R}$ which satisfies $\Lambda^*(f) = f'(0)$ (Since $X_n$ is subspace in $L^p([0,1]))$.
But I am not sure how to extend this result to $X$. Is it enough to show that $X$ is a subspace in $L^p([0,1])$?
Best Answer
Let $(f_n) \subset X$ be given by $$f_n(x) = (x-1)^n$$ so that $\|f_n\|_p = (np+1)^{-1/p} \leq 1$. Let $D_0 f = f^\prime (0)$ be the derivative operator, and note that: $$|D_0f_n| = n \to \infty$$ as $n \to \infty$, which implies that $D_0$ is not bounded on $X$. Thus, $D_0$ cannot be extended to a bounded linear functional on all of $L^p$.