Sumários
15 Dezembro 2025, 11:30
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Maria Teresa Faria da Paz Pereira
Correction of Assignment 6.
Resolution of exercises of Chapter III and IV.
(This class had an extra time of 30mn.)
15 Dezembro 2025, 10:30
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Maria Teresa Faria da Paz Pereira
Alaoglu's Theorem. For E a separable normed space, the weak-star topology in the unitary ball of E' is metrizable.
10 Dezembro 2025, 11:00
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Maria Teresa Faria da Paz Pereira
A convex set of a normed space is closed iff is weakly closed.
The dual of the space c_0.
Weak-* topology in the dual E' of a normed space: definition of the topology; sequences weakly-* convergent. Examples. Properties. Reflexivity and separability. James' theorem (without proof). If E is separable, the unitary closed ball of E' is weakly-* sequencially compact (proof not yet complete).
(This lecture had an extra time of 30mn.)
3 Dezembro 2025, 11:00
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Maria Teresa Faria da Paz Pereira
SThe spaces l^p (1<p<\infty) are reflexive.
Some Riesz Representation Theorems (cont.): for L^p (1<=p<infty, without proof); definition and properties of bounded variation functions; the dual of C([a,b]) with the supremum norm (without proof).
Weak convergence and weak topology on normed spaces. Examples. Properties of weakly convergent sequences.The open unitary ball of an infinitely dimensional normed space is not open for the weak topology.
(Lecture with 30mn of extra time.)
26 Novembro 2025, 11:00
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Maria Teresa Faria da Paz Pereira
Correction of assignment #5. Examples and exercises on normed and Banach spaces (Chapter 3).