## A first course in Hilbert space |

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### Contents

Closed linear subspaces | 23 |

Continuous linear mappings | 35 |

Proper values | 62 |

2 other sections not shown

### Common terms and phrases

1-dimensional approximate proper value Assume Banach algebra Banach space bilinear bounded linear form bounded linear mapping c.c. operator c.c. PROOF Cauchy sequence Cauchy-Schwarz inequality classical Hilbert space clearly closed linear subspace complex numbers complex vector space complex-valued continuous functions convergent sequence COROLLARY defined DEFINITION denote EXAMPLE Exer EXERCISE finite finite-dimensional fn(t following are equivalent following Prop Hint idempotent inequality infinite invariant isometric isomorphism let us show linear combination linearly independent llxll2 llyll non-zero proper value normal and c.c normed space notation null-space one-one orthogonal orthonormal basis orthonormal sequence particular Polarization Identity pre-Hilbert space projection with range proper vector PROPOSITION Remark scalar product self-adjoint sequence of complex sequence xn spectral theorem subsequence Suppose Tx|x Tx|y unitary x|Sy x|xk xll2 xnlyn yll2