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2000 Nov

2000 November 17

Do as many problems as you can. Complete solutions to five problems would be considered a good performance.

  1. a. State the inverse function theorem.\ b. Suppose \(L\colon \mathbb R^3 \rightarrow \mathbb R^3\) is an invertible linear map and that \(g\colon \mathbb R^3 \rightarrow \mathbb R^3\) has continuous first order partial derivatives and satisfies \(\|g(x)\| \leq C\|x\|^2\) for some constant \(C\) and all \(x\in \mathbb R^3\). Here \(\|x\|\) denotes the usual Euclidean norm on \(\mathbb R^3\). Prove that \(f(x) = L(x) + g(x)\) is locally invertible near \(0\).

  2. Let \(f\) be a differentiable real valued function on the interval \((0,1)\), and suppose the derivative of \(f\) is bounded on this interval. Prove the existence of the limit \(L = \lim_{x\rightarrow 0^+} f(x)\).

  3. Let \(f\) and \(g\) be Lebesgue integrable functions on \([0,1]\), and let \(F\) and \(G\) be the integrals

    \[F(x) = \int_0^x f(t) \ dt, \quad G(x) = \int_0^x g(t) \ dt.\]

    Use Fubini's and/or Tonelli's Theorem to prove that

    \[\int_0^1 F(x)g(x) \ dx = F(1) G(1) - \int_0^1 f(x)G(x) \ dx.\]

    Other approaches to this problem are possible, but credit will be given only to solutions based on these theorems.

  4. Let \((X, A, \mu)\) be a finite measure space and suppose \(\nu\) is a finite measure on \((X, A)\) that is absolutely continuous with respect to \(\mu\). Prove that the norm of the Radon-Nikodym derivative \(f = \left[\frac{d\nu}{d\mu}\right]\) is the same in \(L^\infty (\mu)\) as it is in \(L^\infty(\nu)\).

  5. Suppose that \(\{f_n\}\) is a sequence of Lebesgue measurable functions on \([0,1]\) such that \(\lim_{n\rightarrow \infty} \int_0^1 |f_n|\ dx = 0\) and there is an integrable function \(g\) on \([0,1]\) such that \(|f_n|^2 \leq g\), for each \(n\). Prove that \(\lim_n \int_0^1 |f_n|^2\ dx =0\).

  6. Denote by \(\mathcal P_e\) the family of all even polynomials. Thus a polynomial \(p\) belongs to \(\mathcal P_e\) if and only if \(p(x) = \frac{p(x) + p(-x)}{2}\) for all \(x\). Determine, with proof, the closure of \(\mathcal P_e\) in \(L^1[-1,1]\). You may use without proof the fact that continuous functions on \([-1,1]\) are dense in \(L^1[-1,1]\).

  7. Suppose that \(f\) is real valued and integrable with respect to Lebesgue measure \(m\) on \(\mathbb R\) and that there are real numbers \(a<b\) such that

    \[a \cdot m(U) \leq \int_U f \ dm \leq b \cdot m(U),\]

    for all open sets \(U\) in \(\mathbb R\). Prove that \(a \leq f(x) \leq b\) a.e.