Lecture |
Date |
Topics |

25 | 3/17/97 | Historical introduction; norms, inner products, families of seminorms, topological vector spaces; polarization identity and parallelogram law |

26 | 3/19/97 | Examples of topological vector spaces of various sorts |

27 | 3/21/97 | Closed subspaces, quotient spaces, projection onto a closed convex subset of Hilbert space |

28 | 3/24/97 | Example of closed subspaces with a non-closed sum; orthogonal decomposition in Hilbert space; Bessel's inequality |

29 | 3/26/97 | summation over arbitrary index sets; orthonormal bases in Hilbert spaces; Hamel, Hilbert, and Schauder bases |

30 | 3/28/97 | linear operators; completeness of the space of bounded operators; dual spaces; Hahn-Banach Theorem; adjoint operators; annihilators |

31 | 3/31/97 | duals of subspaces and quotient spaces; the Riesz Representation Theorems |

32 | 4/2/97 | duals of function and sequence spaces; biduals and reflexivity; Baire Category Theorem; Open Mapping Theorem |

33 | 4/4/97 | Inverse Mapping Theorem; Closed Graph Theorem; Uniform Boundedness Principle |

34 | 4/7/97 | Weak topology; convex separation theorems; convexity and weak topology |

35 | 4/9/97 | Weak* topology; examples of weak and weak* convergence; Alaoglu's Theorem |

36 | 4/11/97 | more on weak* topology; reflexivity iff unit ball is weakly compact |

37 | 4/14/97 | problem session |

38 | 4/16/97 | Closed Range Theorem; Hilbert-Schmidt operators |

39 | 4/18/97 | Compact operators |

40 | 4/21/97 | Spectral Theorem for compact self-adjoint operators in Hilbert space |

41 | 4/23/97 | Spectral Theorem for compact normal operators in Hilbert space |

42 | 4/25/97 | the spectrum and resolvent of operators in Banach space; preparation for the study of the spectrum of compact operators |

43 | 4/28/97 | the spectrum of a compact operator in a Banach space; Fredholm alternative |

44 | 4/30/97 | general spectral theory; Gelfand-Mazur Theorem; spectral radius formula |

45 | 5/2/97 | Spectral Mapping Theorem; Spectral Theorem for self-adjoint operators in Hilbert space |

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