ENTRY FUNCTIONAL ANALYSIS Authors: Oliver Knill: 2002 Literature: various notes +------------------------------------------------------------ | adjoint +------------------------------------------------------------ The adjoint of a bounded linear operator A on a Hilbert space is the unique operator B which satisfies (Ax,y)=(x,By) for all x,y in H. One calls the adjoint A^*. An bounded linear operator is selfadjoint, if A=A^*. +------------------------------------------------------------ | Alaoglu's theorem +------------------------------------------------------------ Alaoglu's theorem (=Banach-Alaoglu theorem): the closed unit ball in a Banach space is weak-* compact. +------------------------------------------------------------ | angle +------------------------------------------------------------ The angle phi between two vectors v and w of a Hilbert space is a solution phi of the equation cos(phi) ||v|| ||w|| = (v,w), usually the smaller of the two solutions. +------------------------------------------------------------ | balanced +------------------------------------------------------------ A subset Y of a vector space X is called balanced if t x is in Y whenever x is in Y and t<1. +------------------------------------------------------------ | B*-algebra +------------------------------------------------------------ A B*-algebra is a Banach algebra with a conjugate-linear anti-automorphic involution * satisfying ||x x^* || = ||x||^2. +------------------------------------------------------------ | Banach algebra +------------------------------------------------------------ A Banach algebra is an algebra X over the real numbers or complex numbers which is also a Banach space such that ||x y || leq ||x|| ||y|| for all x,y in X. +------------------------------------------------------------ | Banach limit +------------------------------------------------------------ A Banach limit is a translation-invariant functional f on the Banach space of all bounded sequence such that f(c)=c_1 for constant sequences. +------------------------------------------------------------ | Banach space +------------------------------------------------------------ A Banach space is a complete normed space. +------------------------------------------------------------ | barrel +------------------------------------------------------------ A barrel is a closed, convex, absorbing, balanced subset of a topological vector space. +------------------------------------------------------------ | basis +------------------------------------------------------------ A basis (= Schauder basis) of a separable normed space is a sequence of vectors x_j such that every vector x can uniquely be written as y = sum_j y_j x_j. +------------------------------------------------------------ | basis +------------------------------------------------------------ A basis in a vector space is a linearly independent subset that generates the space. +------------------------------------------------------------ | baralled space +------------------------------------------------------------ A baralled space is a topological vector space in which every barrel contains a neighborhood of the origin. +------------------------------------------------------------ | biorthogonal +------------------------------------------------------------ Two sequences a_n and b_n in a Hilbert space are called biorthogonal if A_nm = (a_n,b_m) is an unitary operator. +------------------------------------------------------------ | Bergman space +------------------------------------------------------------ The Bergman space for an open subset G of the complex plane C is the collection of all anlytic function f on G for which intint_G |f(x+iy)|^2 dx dy is finite. It is an example of a Hilbert space. +------------------------------------------------------------ | Buniakovsky inequality +------------------------------------------------------------ The Buniakovsky inequality (=Cauchy-Schwarz inequality) in a Hilbert space tells that |(a,b)| leq ||a|| ||b||. +------------------------------------------------------------ | Cauchy-Schwartz inequality +------------------------------------------------------------ The Cauchy-Schwartz inequality in a Hilbert space H states that |(f,g)| leq ||f|| ||g||. It is also called Buniakovsky inequality or CBS inequality. +------------------------------------------------------------ | compact operator +------------------------------------------------------------ A compact operator is a bounded linear operator A on a Hilbert space, which has the property that the image A(B) of the unit ball B has compact closure in H. +------------------------------------------------------------ | compact operator +------------------------------------------------------------ A bounded operator A on a separable Hilbert space is called diagonalizable if there exists a basis in H such that H v_i = L_i v_i for every basis vector v_i. Compact normal operators are diagonalizable. +------------------------------------------------------------ | dimension +------------------------------------------------------------ The dimension of a Hilbert space H is the cardinality of a basis of H. A Hilbert space is called seperable, if the cardinality of the basis is the cardinality of the integers. +------------------------------------------------------------ | Egorov's theorem +------------------------------------------------------------ Egorov's theorem Let (X,S,m) be a measure space, where m(S) has finite measure. If a sequence f_n of measurable functions converges to f almost everywhere, then for every d>0, there is a set E_d subset X such that f_n to f uniformly on E setminus E_d and m(E_d)0, there is a set E_d with m(E_d)