Lecture Notes in Functional Analysis

by William L. Paschke

edition 0.9




I. Hilbert Space

II. Bounded Operators

III. Compact Operators

IV. The Spectral Theorem

Index and References


Hilbert space  
Hilbert space tensor product  HHilbert space(complete) tensor productKHilbert space\( H\mathbin{\hat{\otimes}} K \)
Hilbert-Schmidt operator  
adjoint  Tlinear map*adjoint\( T^{*} \)
algebraic tensor product  Vvector space1onetensor producttensor productVvector spaceninteger\( {V}_{1}\otimes \dotsb\otimes {V}_{n} \)
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  Usubspacedirect sumWsubspace\( U\oplus W \)
distance to subset  ddistance\( \operatorname{d} \)
dual space  Xnormed linear space*dual space\( {X}^{*} \)
essential supremum  L-infinity normφbounded measurable functionL-infinity norm\( \mathopen{}\left\lVert{}φ\right\rVert_\infty\mathclose{} \)
external direct sum  HHilbert space1onedirect sumdirect sumHHilbert spaceninteger\( {H}_{1}\oplus \dotsb\oplus {H}_{n} \)
graph  Ggraph\( \mathop{\mathcal{G}} \)
inner product  inner product·, ·inner product\( \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{} \)
inner product space  
integral operator  
isomorphism  HHilbert spaceisomorphicKHilbert space\( H\simeq K \)
norm  norm·norm\( \mathopen{}\left\lVert{}\cdot\right\rVert\mathclose{} \)
normal operator  
normed algebra  
normed linear space  
null vector  
orthogonal  xvectororthogonalyvector\( x\perp y \)
orthogonal complement  Ssubsetorthogonal complement\( {S}^{\perp} \)
orthogonal complement in subspace  Tclosed subspacedirect minusTclosed subspace1one\( T\ominus {T}_{1} \)
orthogonal projection  
orthonormal basis  
orthogonal projection operator  
p-norm  p-normfcontinuous functionpp-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\( \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{} \)
quotient space  Vvector space/modSsubspace\( V/S \)
self-adjoint operator  
simple tensors  
spectral radius  ρspectral radius\( \mathop{\rho} \)
spectral resolution  
spectrum  σspectrum\( \mathop{\sigma} \)
supremum norm  supremum normfcontinuous functionsupremum norm\( \mathopen{}\left\lVert{}f\right\rVert_\infty\mathclose{} \)
weak*-topology  w* weak*-topology \( \mathop{\mathrm{w}^*} \)


[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.