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2001 Nov

2001 November 26

Instructions. Masters students do any 4 problems. Ph.D. students do any 5 problems. Use a separate sheet of paper for each new problem.

  1. Let \(\{f_n\}\) be a sequence of Lebesgue measurable functions on a set \(E\subset \mathbb R\), where \(E\) is of finite Lebesgue measure. Suppose that there is \(M>0\) such that \(|f_n(x)|\leq M\) for \(n\geq 1\) and for all \(x \in E\), and suppose that \(\lim_n f_n(x) = f(x)\) for each \(x\in E\). Use Egoroff's Theorem to prove that

    \[\int_E f(x)\ dx = \lim_n \int_E f_n(x)\ dx.\]

  2. Let \(f(x)\) be a real-valued Lebesgue integrable function on \([0,1]\).\ a. Prove that if \(f>0\) on a set \(F\subset [0,1]\) of positive measure, then \(\int_F f(x)\ dx > 0.\)\ b. Prove that if \(\int_0^x f(x)\ dx =0\) for each \(x\in [0,1]\), then \(f(x)=0\) for almost all \(x\in [0,1]\).

  3. State each of the following:\ a. The Stone-Weierstrass theorem\ b. The Lebesgue (dominated) convergence theorem\ c. Holder's inequality\ d. The Riesz representation theorem for \(L^p\)\ e. The Hahn-Banach theorem.

  4. a. State the Baire category theorem.\ b. Prove the following special case of the uniform boundedness theorem: Let \(X\) be a (nonempty) complete metric space and let \(F\subseteq C(X)\). Suppose that for each \(x\in X\) there is a nonnegative constant \(M_x\) such that

    \[|f(x)| \leq M_x \quad \text{ for all } \quad f\in F.\]

    Prove that there is a nonempty \emph{open} set \(G\subseteq X\) and a constant \(M>0\) such that \(|f(x)| \leq M\) holds for all \(x\in G\) and for all \(f\in F\).

  5. Prove or disprove:\ a. \(L^2\) convergence implies pointwise convergence.\ b. \(\lim_n \int_0^\infty \frac{\sin(x^n)}{x^n}\ dx = 0\).\ c. Let \(\{f_n\}\) be a sequence of measurable functions defined on \([0,\infty)\). If \(f_n\rightarrow 0\) uniformly on \([0,\infty)\), as \(n\rightarrow \infty\), then

    \[\varliminf \int_{[0,\infty)} f_n(x)\ dx = \int_{[0,\infty)} \varliminf f_n(x)\ dx.\]

  6. Let \(f \colon H\to H\) be a bounded linear functional on a separable Hilbert space H (with inner product denoted by \(\langle \cdot, \cdot \rangle\)). Prove that there is a unique element \(y\in H\) such that

    \[f(x) = \langle x, y \rangle \quad \text{for all} \quad x\in H \quad \text{and} \quad \|f\| = \|y\|.\]

    (Hint. You may use the following facts: A separable Hilbert space, \(H\), contains a complete orthonormal sequence, \(\{\phi_k\}_{k=1}^\infty\), satisfying the following properties:\ i. If \(x,y\in H\) and if \(\langle x,\phi_k \rangle = \langle y,\phi_k \rangle\) for all \(k\), then \(x=y\).\ ii. Parseval's equality holds; that is, for all \(x\in H\), \(\langle x, x \rangle = \sum_{k=1}^\infty a_k^2\), where \(a_k = \langle x,\phi_k \rangle\).

  7. Let \(X\) be a normed linear space and let \(Y\) be a Banach space. Let \(B(X,Y) = \{A : A\colon X\to Y \text{ is a bounded linear operator}\}\). Then with the norm \(\|A\| = \sup_{\|x\|\leq 1} \|Ax\|\), \(B(X,Y)\) is a normed linear space (you need not show this). Prove that \(B(X,Y)\) is a Banach space; that is, prove that \(B(X,Y)\) is complete.