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Last updated:

December 24, 2022


Unlimited Duration


This course includes:

Unlimited Duration

Badge on Completion

Certificate of completion

Unlimited Duration


This is a undergraduate course

It will cover normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators; Lebesgue measure, measurable functions, integrability, completeness of L-p spaces; Hilbert space; compact, Hilbert-Schmidt and trace class operators; as well as spectral theorem.

Course Curriculum

  • Linear spaces, metric spaces, normed spaces Unlimited
  • Linear maps between normed spaces Unlimited
  • Banach spaces Unlimited
  • Lebesgue integrability Unlimited
  • Lebesgue integrable functions form a linear space Unlimited
  • Null functions Unlimited
  • Monotonicity, Fatou’s Lemma and Lebesgue dominated convergence Unlimited
  • Hilbert spaces Unlimited
  • Baire’s theorem and an application Unlimited
  • Bessel’s inequality Unlimited
  • Closed convex sets and minimizing length Unlimited
  • Compact sets. Weak convergence. Weak compactness Unlimited
  • Baire’s theorem. Uniform boundedness. Boundedness of weakly convergent sequences Unlimited
  • Fourier series and L2 Unlimited
  • Open mapping and closed graph theorems Unlimited
  • Bounded operators. Unitary operators. Finite rank operators Unlimited
  • The second test Unlimited
  • Compact operators Unlimited
  • Fredholm operators Unlimited
  • Completeness of the eigenfunctions Unlimited
  • Dirichlet problem for a real potential on an interval Unlimited
  • Dirichlet problem (cont.) Unlimited
  • Harmonic oscillator Unlimited
  • Completeness of Hermite basis Unlimited
  • The fourier transform on the line Unlimited
  • Hahn-Banach and review Unlimited

About the instructor

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Massachusetts Institute of Technology
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