## A First Course in Functional Analysis"A First Course in Functional Analysis lucidly covers Banach Spaces. Continuous linear functionals, the basic theorems of bounded linear operators, Hilbert spaces, Operators on Hilbert spaces. Spectral theory and Banach Algebras usually taught as a core course to post-graduate students in mathematics. The special distinguishing features of the book include the establishment of the spectral theorem for the compact normal operators in the infinite dimensional case exactly in the same form as in the finite dimensional case and a detailed treatment of the theory of Banach algebras leading to the proof of the Gelfand-Neumark structure theorem for Banach algebras."--BOOK JACKET. |

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### Contents

Banach Spaces | 1 |

Continuous Linear Functionals | 79 |

The Basic Theorems of Bounded Linear Operators | 140 |

5 other sections not shown

### Common terms and phrases

adjoint operation B*-algebra Banach algebra Banach space basis Cauchy sequence closed linear subspace closed subspace compact operator complete orthonormal set completes the proof complex numbers conjugate space continuous linear functionals contradicting convergent subsequence COROLLARY countable definition denoted dense e a(T eigenvalue eigenvector element EXAMPLE following theorem given Hahn-Banach theorem Hence Hilbert space homeomorphism hypothesis identity implies inequality inner product space invertible isometric isomorphism Lemma let us assume Let x e linear subspace linear transformation ll/ll llxll llyll matrix maximal ideal metric space natural imbedding non-empty non-zero vector normal operator normed linear space NOTE one-to-one open mapping theorem open set open sphere operator on H orthogonal Pa(T positive integer projection on H PROOF Let properties real numbers reflexive self-adjoint operators space H spectral subset subspace of H topology unique x e H xv x2