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Math Help - not continuous linear functional

  1. #1
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    not continuous linear functional

    Hi again! We are in a L^p-space with 0<p<1 ( L^p=\{f:[0,1] \rightarrow \mathbb{K}: f L-measurable,  \int_0^1 |f(t)|^p dt < \infty\} with d(f,g):= \int_0^1 |f(t)-g(t)|^p dt.
    We consider a linear functional \phi \neq 0. Let f \in L^p: \int |f|^p =1, \phi(f)=\alpha > 0. Let F(t):=\int_0^t |f(s)|^p ds. Then F is continuous, therefore attains all values \frac{k}{n}, say at s_k, k=0,...,n. Define g_r:=f \cdot c_{[s_{r-1},s_r]} (c meaning the char. function). There is a r(n), so that |\phi(g_{r(n)})| \geq \frac{\alpha}{n}. Using f_n:=n \cdot g_r(n) one can now show that \phi is not continuous. But how?
    I do not see why that follows. Does anybody know and can help me?
    Last edited by Recursion; January 13th 2009 at 01:46 PM.
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