Tag Archives: Quals

The Sphere is Weakly Dense in the Ball

Another qual question, this one dealing with Hilbert spaces and the weak topology: Exercise: Let \(\mathcal{H}\) be an infinite dimensional Hilbert space. Show that the unit sphere \(S := \{x\in\mathcal{H} : \|x\| = 1\}\) is weakly dense in the unit … Continue reading

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Nowhere Differentiable Functions

In an undergraduate analysis class, one of the first results that is generally proved after the definition of differentiability is given is the fact that differentiable functions are continuous. We can justifiably ask if the converse holds—are there examples of … Continue reading

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An Elementary Duality Result

The following problem is a fairly straightforward exercise that seems to come up fairly often on qualifying exams at UCR, and I think that I finally have a nicely argued proof. I am going to put it up here mostly … Continue reading

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