Exam
6710-2022-08
Fall 2022 · 5 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Let \((u_n)\) and \((v_n)\) be two orthonormal sequences in a Hilbert space \(H\). Assume that
\[\sum _{n=1}^{\infty } \|u_n - v_n\|^2 < 1.\]Show that if \((u_n)\) is a total orthonormal sequence, then \((v_n)\) is also a total orthonormal sequence. Hint: Assume for contradiction that \((v_n)\) is not total.
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Question 2
Let \(X\) be a real Banach space. Assume \(f \in X^*\) has a closed nullspace \(\mathcal {N}(f)\). The goal of this problem is to show that \(f\) must be a bounded linear functional.
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Let \(x_0 \in X\) be such that \(f(x_0) = 1\). Explain why there is an \(\epsilon > 0\) for which the ball
\[B(x_0, \epsilon ) \equiv \{x \in X \mid \|x - x_0\| < \epsilon \}\]satisfies \(B(x_0, \epsilon ) \subset X - \mathcal {N}(f)\).
- Prove that \(f(x) > 0\) for all \(x \in B(x_0, \epsilon )\), where \(\epsilon \) is as in part (a). To prove this, you may assume for contradiction that there is some \(y \in B(x_0, \epsilon )\) with \(f(y) \le 0\).
- Any \(x \in B(x_0, \epsilon )\) can be written as \(x = x_0 + \epsilon u\) for some \(u\) with \(\|u\| < 1\). Use the result of part (b) to show that \(|f(u)| < 1/\epsilon \).
- To conclude, give an upper bound for \(\|f\|\).
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Question 3
Let \(H\) be a separable Hilbert space with \((e_n)\) being a total orthonormal sequence of \(H\). Let \(T : H \to H\) be a bounded linear operator satisfying
\[\sum _{n=1}^{\infty } \|T e_n\|^2 < \infty .\]Show that \(T\) is compact by approximating \(T\) by a sequence \((T_k)\) of finite rank operators that converges to \(T\) in an appropriate sense.
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Question 4
Let \(X\) be a non-trivial normed vector space. Use the Hahn-Banach theorem to show that for any \(x \in X\) we have that
\[\|x\| = \sup _{f \in X', f \neq 0} \frac {|f(x)|}{\|f\|}.\] -
Question 5
Consider the left shift operator \(L : \ell ^2 \to \ell ^2\), defined for a sequence \((x_n) \in \ell ^2\) by
\[L(x_1, x_2, x_3, \ldots ) = (x_2, x_3, x_4, \ldots ).\]- Prove that \(\|L\| = 1\).
- Explain why if \(\lambda \in \mathbb {C}\) with \(|\lambda | > 1\), then \(\lambda \in \rho (L)\), the resolvent set of \(L\).
- Prove that that if \(\lambda \in \mathbb {C}\) with \(|\lambda | < 1\), then \(\lambda \in \sigma _p(L)\), the point spectrum of \(L\).
- Deduce what is \(\sigma (L)\), the spectrum of \(L\).