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\title{ HW IV for Math 6210}
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1) Let $H$ be a Hilbert space and $P:H \rightarrow H$ a continuous operator such that $P^2=P$ and $P^*=P$. Prove that the image of $P$ is closed and that 
 $H$ is isomorphic to the direct sum of the kernel of $P$ and the image of $P$. 

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2) Let $H$ be a Hilbert space and $e_1$, $e_2, \ldots$ and an orthonormal basis. Prove that $0$ is a weak limit of the sequence $e_n$. Prove that the unit ball $B$, 
$||x|| \leq 1$, is closed in the weak topology. Prove that the unit sphere $S$, $||x||=1$, is dense in $B$ in the weak topology. 

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3) Let $(X,\mathcal M, \mu)$ and $(Y, \mathcal N, \nu)$ be two $\sigma$-finite measure spaces. If functions $e_1, e_2 \ldots $ are a Hilbert space basis of 
$L^2(X)$ and functions $f_1, f_2, \ldots $ are a Hilbert space basis of $L^2(Y)$, prove that $e_i\cdot f_j$ is a basis of $L^2(X\times Y)$. 
(Here $e_i\cdot f_j$ is a function on $X\times Y$ such that its value at $(x,y)\in X\times Y$ is $e_i(x)\cdot f_j(y)$.) 



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4) Let $X=[0,1]$ with the usual measure.  Let $p\geq 1$. For $n=1,2, \ldots$, let $f_n=n^{1/p} \cdot \chi_{[0,1/n]}$. Prove that the sequence $f_n$ converges weakly to 0 
in $L^p([0,1)]$ if and only if $p>1$. (Use that $L^p([0,1)]^* \cong L^q([0,1)]$.) 


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5) Let $1 < p < q < \infty$. Then $L^p$ is not contained in $L^q$ iff $X$ contains sets of arbitrarily small positive measure, and $L^q$ is not contained in 
$L^p$ iff $X$ contains sets of arbitrarily large finite measure. This is exercise 5, page 187, see the hints there. 

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6)  Exercise 18 on page 191.  Follow the hints there to prove the Radon-Nikodym Theorem for a finite measure space $(X,\mathcal M,\mu)$, using $L^2(X)^*\cong L^2(X)$ which is 
a part of the Hilbert space theory. (As usually, for simplicity assume we deal with real valued functions.)  


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4) Let $X$ be a measure space such that $\mu(X)=1$. Let $p>1$. Let $f_n$ be a sequence in $L^p(X)$ such that $f_n(x) \rightarrow 0$ for almost all $x$ and 
$|| f_n||_p \leq 1$. Prove that the sequence $f_n$ converges weakly to 0. Hint: this is a generalization of the previous exercise. 
Use the theorem of Egoroff, page 62, and Theorem 3.5, page 89, for the measure $\nu(E)=\int_E |g|^q$ where $g\in L^q(X)$. 




