Lecture Notes in Functional Analysis

by William L. Paschke

edition 0.9

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Contents

Frontmatter

I. Hilbert Space

II. Bounded Operators

III. Compact Operators

IV. The Spectral Theorem

Index and References

Index

l2little l 2\( \mathrm{l}^{0} \)
l0little l 0\( \)
Banach space  
Calkin algebra  
Cauchy sequence  
Hilbert space  
Hilbert space tensor product  HHilbert space⨶(complete) tensor productKHilbert space\( H\mathbin{\hat{\otimes}} K \)
Hilbert-Schmidt operator  
Volterra Operator  
adjoint  Tlinear map*adjoint\( T^{*} \)
algebraic tensor product  Vvector space1one⊗tensor product⋯⊗tensor productVvector spaceninteger\( {V}_{1}\otimes \dotsb\otimes {V}_{n} \)
bounded linear operator  
closable linear operator  
closed linear operator  
closure of linear operator  Llinear operator¯norm closure\( \overline{L} \)
compact operator  
complete metric space  
complex group algebra  Ccomplex group algebraGgroup\( \mathbb{C}G \)
conjugate linear  
cyclic subspace  
cyclic vector  
complex unit circle  Tcomplex unit circle\( \mathbb{T} \)
direct sum  Usubspace⊕direct sumWsubspace\( U\oplus W \)
distance to subset  ddistance\( \operatorname{d} \)
dual space  Xnormed linear space*dual space\( {X}^{*} \)
essential supremum  ‖L-infinity normφbounded measurable function‖∞L-infinity norm\( \mathopen{}\left\lVert{}φ\right\rVert_\infty\mathclose{} \)
extension  
external direct sum  HHilbert space1one⊕direct sum⋯⊕direct sumHHilbert spaceninteger\( {H}_{1}\oplus \dotsb\oplus {H}_{n} \)
finite rank  
graph  Ggraph\( \mathop{\mathcal{G}} \)
inner product  〈inner product·, ·〉inner product\( \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{} \)
inner product space  
integral operator  
isometry  
isomorphism  HHilbert space≃isomorphicKHilbert space\( H\simeq K \)
metric  
metric space  
multilinear  
multiplication operator  
norm  ‖norm·‖norm\( \mathopen{}\left\lVert{}\cdot\right\rVert\mathclose{} \)
normal operator  
normed algebra  
normed linear space  
null vector  
open  
orthogonal  xvector⊥orthogonalyvector\( x\perp y \)
orthogonal complement  Ssubset⊥orthogonal complement\( {S}^{\perp} \)
orthogonal complement in subspace  Tclosed subspace⊖direct minusTclosed subspace1one\( T\ominus {T}_{1} \)
orthogonal projection  
orthonormal  
orthonormal basis  
orthogonal projection operator  
p-norm  ‖p-normfcontinuous function‖pp-norm\( \mathopen{}\left\lVert{}f\right\rVert_p\mathclose{} \)
partial isometry  
positive operator  
positive definite matrix  
positive definite function  
positive sesquilinear form  〈positive sesquilinear form·, ·〉positive sesquilinear form\( \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{} \)
projection  
quotient space  Vvector space/modSsubspace\( V/S \)
resolution of the identity  
self-adjoint operator  
simple tensors  
spectral radius  ρspectral radius\( \mathop{\rho} \)
spectral resolution  
spectrum  σspectrum\( \mathop{\sigma} \)
supremum norm  ‖supremum normfcontinuous function‖∞supremum norm\( \mathopen{}\left\lVert{}f\right\rVert_\infty\mathclose{} \)
trace-class operator  
ultraweak topology  
unitary operator  
weak*-topology  w* weak*-topology \( \mathop{\mathrm{w}^*} \)

References

[1]Bachman, George; Narici, Lawrence; Functional analysis. Academic Press, New York. 1966.

[2]Friedman, Avner; Foundations of modern analysis. Holt, Rinehart and Winston, Inc., New York. 1970.

[3]Gohberg, Israel; Goldberg, Seymour; Krupnik, Nahum; Traces and determinants of linear operators. Birkhäuser Verlag, Basel. 2000.

[4]Istrăţescu, Vasile Ion; Inner product structures. D. Reidel Publishing Co., Dordrecht. 1987.

[5]Kadison, Richard V.; Ringrose, John R.; Fundamentals of the theory of operator algebras. Vol. I. American Mathematical Society, Providence, RI. 1997.

[6]Kadison, Richard V.; Ringrose, John R.; Fundamentals of the theory of operator algebras. Vol. II. American Mathematical Society, Providence, RI. 1997.

[7]Köthe, Gottfried; Topological vector spaces. I. Springer-Verlag New York, Inc., New York. 1969.

[8]Murphy, Gerard J.; C*-algebras and operator theory. Academic Press, Inc., Boston, MA. 1990.

[9]Rickart, Charles E.; General theory of Banach algebras. D. Van Nostrand Co., Inc., Princeton, N.J.. .

[10]Rudin, Walter; Principles of mathematical analysis. McGraw-Hill Book Co., New York. 1976.

[11]Yosida, Kôsaku; Functional analysis. Springer-Verlag, New York. 1974.

[12]Young, Nicholas; An introduction to Hilbert space. Cambridge University Press, Cambridge. 1988.