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The function $r_{n}:[0,1]\rightarrow \{-1,1\}$ be defined by $r_{n}(t)=\operatorname{sgn}(\sin(2^{n}\pi t))$ is known as the $n$-th Rademacher function.

a) Show that $r_{n}\xrightarrow{w}0$ in $L_1$;

b) Does $r_{n}\xrightarrow{w}0$ in $L_p$ for $1<p<\infty$?

c) Show that in $L_{\infty}$, the sequence $(r_n)$ spans an isometric copy of $\ell_1$.

I'm really stuck and I don't know how to begin, any ideas or hints please. Thank you.

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(a) what does it mean to converge weakly in $L^1$? (b) what does it mean to converge weakly in $L^p$? (c) what is the $L^\infty$ norm of a linear combination of $r_n$? – user53153 Feb 5 '13 at 18:37
Hint for c): The isometry is given by $e_i\mapsto r_i$. Look at the first $n$ Rademachers in $L_\infty$ and note that given any sequence of signs $(\epsilon_1,\ldots,\epsilon_n)$, there is an interval $I$ of length $2^{1-n}$ so that $r_j(x)=\epsilon_j$ for each $j=1,\ldots, n$ and each $x\in I$. – David Mitra Feb 5 '13 at 18:41
@5PM a) means for all linear functionals $f\in (L_1)^{*}$ we have $f(r_n)\rightarrow 0$ similarly for b) – i.a.m Feb 5 '13 at 18:43
@i.a.m Actually, I wrote that comment trying to hint "where to begin": you should begin by writing out the notion of weak convergence - not by using dual space formalism, but in terms of actual integrals $\int_0^1 r_n g$ where $g\in L^\infty$ or $g\in L^q$ with $1/p+1/q=1$. – user53153 Feb 5 '13 at 18:48
@5PM ah, I see thatn you for the hint I'll try to do it – i.a.m Feb 5 '13 at 18:58

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