Sumários
Lecture 15
8 Novembro 2023, 11:00 • Maria Teresa Faria da Paz Pereira
Riesz's Lemma; consequences.
Hahn-Banach Theorem - real version (proof not yet complete).
Lecture 14
6 Novembro 2023, 10:30 • Maria Teresa Faria da Paz Pereira
Linear hyperplanes in normed spaces. relation with linear functionals. A linear functional f in a normed space is continuos iff Ker f is closed. Examples and counter-examples.
Zorn's Lemma (revision). Vector spaces of infinite dimension: Hamel bases; Schauder bases for normed spaces; examples.
Lecture 13
30 Outubro 2023, 10:30 • Maria Teresa Faria da Paz Pereira
Still some consequences of the theorem of Spectral decomposition for compact self-adjoint operators T: ad(Im T) has a Hilbert basis; the spectrum of T coincides with the point spectrum with the possible union with the constant 0.
Banach spaces: examples; the spaces c_0 and c of l^\infty are closed. Quotient spaces E/M, for E a normed space and M a closed linear subspace: definition; norm in E/M; first properties; the canonical map and its properties.
TP 6
25 Outubro 2023, 11:30 • Maria Teresa Faria da Paz Pereira
Examples and resolution of exercises of Chapter I.