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t 102 overall von neumann s major achievement in set theory was an axiomatization of set theory and connected with that elegant theory of the ordinal and cardinal numbers as well as the first strict formulation of principles of definitions by the transfinite induction 104 von neumann paradox edit main article von neumann paradox building on the work of felix hausdorff in 1924 stefan banach and alfred tarski proved that given a solid ball in 3 dimensional space there exists a decomposition of the ball into a finite number of disjoint subsets that can be reassembled together in a different way to yield two identical copies of the original ball banach and tarski proved that using isometric transformations the result of taking apart and reassembling a two dimensional figure would necessarily have the same area as the original this would make creating two unit squares out of one impossible but in a 1929 paper 105 von neumann proved that paradoxical decompositions could use a group of transformations that include as a subgroup a free group with two generators the group of area preserving transformations contains such subgroups and this opens the possibility of performing paradoxical decompositions using these subgroups the class of groups von neumann isolated in his work on banach tarski decompositions was very important in many areas of mathematics including von neumann s own later work in measure theory see below proof theory edit see also hilbert s program with the aforementioned contributions of von neumann to sets the axiomatic system of the theory of sets avoided the contradictions of earlier systems and became usable as a foundation for mathematics despite the lack of a proof of its consistency the next question was whether it provided definitive answers to all mathematical questions that could be posed in it or whether it might be improved by adding stronger axioms that could be used to prove a broader class of theorems by 1925 he was involving himself in discussions with others in göttingen on whether elementary arithmetic followed from peano axioms 106 building on the work of ackermann von neumann began attempting to prove using the finistic methods of hilbert s school the consistency of first order arithmetic he succeeded in proving the consistency of a fragment of arithmetic of natural numbers through the use of restrictions on induction 107 he continued looking for a more general proof of the consistency of classical mathematics using methods from proof theory 108 a strongly negative answer to whether it was definitive arrived in september 1930 at the historic second conference on the epistemology of the exact sciences of königsberg in which kurt gödel announced his first theorem of incompleteness the usual axiomatic systems are incomplete in the sense that they cannot prove every truth expressible in their language moreover every consistent extension of these systems necessarily remains incomplete 109 less than a month later von neumann who had participated in the conference communicated to gödel an interesting consequence of his theorem that the usual axiomatic systems are unable to demonstrate their own consistency 109 gödel had already discovered this consequence now known as his second incompleteness theorem and sent von neumann a preprint of his article containing both theorems 110 von neumann acknowledged gödel s priority in his next letter 111 he never thought much of the american system of claiming personal priority for everything 112 however von neumann s method of proof differed from gödel s as his used polynomials to explain consistency 113 114 with this discovery von neumann ceased work in mathematical logic and foundations of mathematics and instead spent time on problems connected with applications 115 ergodic theory edit in a series of papers published in 1932 von neumann made foundational contributions to ergodic theory a branch of mathematics that involves the states of dynamical systems with an invariant measure 116 of the 1932 papers on ergodic theory paul halmos wrote that even if von neumann had never done anything else they would have been sufficient to guarantee him mathematical immortality 117 by then von neumann had already written his articles on operator theory and the application of this work was instrumental in his mean ergodic theorem 118 the theorem is about arbitrary one parameter unitary groups t v t displaystyle mathit t to mathit v_ t and states that for every vector ϕ displaystyle phi in the hilbert space lim t 1 t 0 t v t ϕ d t displaystyle lim _ t to infty frac 1 t int _ 0 t v_ t phi dt exists in the sense of the metric defined by the hilbert norm and is a vector ψ displaystyle psi which is such that v t ψ ψ displaystyle v_ t psi psi for all t displaystyle t this was proven in the first paper in the second paper von neumann argued that his results here were sufficient for physical applications relating to boltzmann s ergodic hypothesis he also pointed out that ergodicity had not yet been achieved and isolated this for future work 119 later in the year he published another long and influential paper that began the systematic study of ergodicity in this paper he gave and proved a decomposition theorem showing that the ergodic measure preserving actions of the real line are the fundamental building blocks from which all measure preserving actions can be built several other key theorems are given and proven the results in this paper and another in conjunction with paul halmos have significant applications in other areas of mathematics 120 121 measure theory edit see also lifting theory in measure theory the problem of measure for an n dimensional euclidean space r n may be stated as does there exist a positive normalized invariant and additive set function on the class of all subsets of r n 122 the work of felix hausdorff and stefan banach had implied that the problem of measure has a positive solution if n 1 or n 2 and a negative solution because of the banach tarski paradox in all other cases von neumann s work argued that the problem is essentially group theoretic in character the existence of a measure could be determined by looking at the properties of the transformation group of the given space the positive solution for spaces of dimension at most two and the negative solution for higher dimensions comes from the fact that the euclidean group is a solvable group for dimension at most two and is not solvable for higher dimensions thus according to von neumann it is the change of group that makes a difference not the change of space 123 around 1942 he told dorothy maharam how to prove that every complete σ finite measure space has a multiplicative lifting however he did not publish this proof and she later came up with a new one 124 in a number of von neumann s papers the methods of argument he employed are considered even more significant than the results in anticipation of his later study of dimension theory in algebras of operators von neumann used results on equivalence by finite decomposition and reformulated the problem of measure in terms of functions 125 a major contribution von neumann made to measure theory was the result of a paper written to answer a question of haar regarding whether there existed an algebra of all bounded functions on the real number line such that they form a complete system of representatives of the classes of almost everywhere equal measurable bounded functions 126 he proved this in the positive and in later papers with stone discussed various generalizations and algebraic aspects of this problem 127 he also proved by new methods the existence of disintegrations for various general types of measures von neumann also gave a new proof on the uniqueness of haar measures by using the mean values of functions although this method only worked for compact groups 126 he had to create entirely new techniques to apply this to locally compact groups 128 he also gave a new ingenious proof for the radon nikodym theorem 129 his lecture notes on measure theory at the institute for advanced study were an important source for knowledge on the topic in america at the time and were later published 130 131 132 topological groups edit using his previous work on measure theory von neumann made several contributions to the theory of topological groups beginning with a paper on almost periodic functions on groups where von neumann extended bohr s theory of almost periodic functions to arbitrary groups 133 he continued this work with another paper in conjunction with bochner that improved the theory of almost periodicity to include functions that took on elements of linear spaces as values rather than numbers 134 in 1938 he was awarded the bôcher memorial prize for his work in analysis in relation to these papers 135 136 in a 1933 paper he used the newly discovered haar measure in the solution of hilbert s fifth problem for the case of compact groups 137 the basic idea behind this was discovered several years earlier when von neumann published a paper on the analytic properties of groups of linear transformations and found that closed subgroups of a general linear group are lie groups 138 this was later extended by cartan to arbitrary lie groups in the form of the closed subgroup theorem 139 126 functional analysis edit main article operator theory see also spectral theorem von neumann was the first person to axiomatically define an abstract hilbert space whereas it was previously defined as the lp space he defined it as a complex vector space with a hermitian scalar product with the corresponding norm being both separable and complete in the same papers he also defined several other abstract inequalities such as the cauchy schwarz inequality that were previously only defined for euclidean spaces 140 he continued with the development of the spectral theory of operators in hilbert space in 3 seminal papers between 1929 and 1932 141 this work cumulated in his mathematical foundations of quantum mechanics which among two other books by stone and banach in the same year were the first monographs on hilbert space theory 142 previous work by others showed that a theory of weak topologies could not be obtained by using sequences and von neumann was the first to outline a program of how to overcome the difficulties which resulted in him defining locally convex spaces and topological vector spaces for the first time in addition several other topological properties he defined at the time he was among the first mathematicians to apply new topological ideas from hausdorff from euclidean to hilbert spaces 143 such as boundness and total boundness are still used today 144 for twenty years von neumann was considered the undisputed master of this area 126 these developments were primarily prompted by needs in quantum mechanics where von neumann realized the need to extend the spectral theory of hermitian operators from the bounded to the unbounded case 145 other major achievements in these papers include a complete elucidation of spectral theory for normal operators the first abstract presentation of the trace of a positive operator 146 147 a generalisation of riesz s presentation of hilbert s spectral theorems at the time and the discovery of hermitian operators in a hilbert space as distinct from self adjoint operators which enabled him to give a description of all hermitian operators which extend a given hermitian operator in addition he wrote a paper detailing how the usage of infinite matrices common at the time in spectral theory was inadequate as a representation for hermitian operators his work on operator theory lead to his most profound invention in pure mathematics the study of von neumann algebras and in general of operator algebras 148 his later work on rings of operators lead to him revisiting his earlier work on spectral theory and providing a new way of working through the geometric content of the spectral theory by the use of direct integrals of hilbert spaces 149 like in his work on measure theory he proved several theorems that he did not find time to publish nachman aronszajn and k t smith were told by him that in the early 1930s he proved the existence of proper invariant subspaces for completely continuous operators in a hilbert space while working on the invariant subspace problem 150 with i j schoenberg he wrote several items investigating translation invariant hilbertian metrics on the real number line which resulted in their complete classification their motivation lie in various questions related to embedding metric spaces into hilbert spaces 151 152 with pascual jordan he wrote a short paper giving the first derivation of a given norm from an inner product by means of the parallelogram identity 153 his trace inequality is a key result of matrix theory used in matrix approximation problems 154 he also first presented the idea that the dual of a pre norm is a norm in the first major paper discussing the theory of unitarily invariant norms and symmetric gauge functions now known as symmetric absolute norms 155 156 157 this paper leads naturally to the study of symmetric operator ideals and is the beginning point for modern studies of symmetric operator spaces 158 later with robert schatten he initiated the study of nuclear operators on hilbert spaces 159 160 tensor products of banach spaces 161 introduced and studied trace class operators 146 their ideals and their duality with compact operators and preduality with bounded operators 162 the generalization of this topic to the study of nuclear operators on banach spaces was among the first achievements of alexander grothendieck 163 164 previously in 1937 von neumann published several results in this area for example giving 1 parameter scale of different cross norms on l 2 n l 2 n displaystyle textit l _ 2 n otimes textit l _ 2 n and proving several other results on what are now known as schatten von neumann ideals 165 operator algebras edit main article von neumann algebra see also direct integral von neumann founded the study of rings of operators through the von neumann algebras originally called w algebras while his original ideas for rings of operators existed already in 1930 he did not begin studying them in depth until he met f j murray several years later 166 167 a von neumann algebra is a algebra of bounded operators on a hilbert space that is closed in the weak operator topology and contains the identity operator 168 the von neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as being equal to the bicommutant 169 after elucidating the study of the commutative algebra case von neumann embarked in 1936 with the partial collaboration of murray on the noncommutative case the general study of factors classification of von neumann algebras the six major papers in which he developed that theory between 1936 and 1940 rank among the masterpie...
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