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Let X and Y be Banach spaces and let f:X->Y be a function between them. f is said to be Gâteaux differentiable if there exists an operator T_x:X->Y such that, for all v in X, ...

The tilde is the mark "~" placed on top of a symbol to indicate some special property. x^~ is voiced "x-tilde." The tilde symbol is commonly used to denote an operator. In ...

The phrase Tomita-Takesaki theory refers to a specific collection of results proven within the field of functional analysis regarding the theory of modular Hilbert algebras ...

If A=>B and B=>A (i.e., A=>B ^ B=>A, where => denotes implies), then A and B are said to be equivalent, a relationship which is written symbolically in this work as A=B. The ...

A statement which appears self-contradictory or contrary to expectations, also known as an antinomy. Curry (1977, p. 5) uses the term pseudoparadox to describe an apparent ...

The notion of a Hilbert C^*-module is a generalization of the notion of a Hilbert space. The first use of such objects was made by Kaplansky (1953). The research on Hilbert ...

Reverse Polish notation (RPN) is a method for representing expressions in which the operator symbol is placed after the arguments being operated on. Polish notation, in which ...

Given a Hilbert space H, a *-subalgebra A of B(H) is said to be a von Neumann algebra in H provided that A is equal to its bicommutant A^('') (Dixmier 1981). Here, B(H) ...

The axiom of Zermelo-Fraenkel set theory which asserts that sets formed by the same elements are equal, forall x(x in a=x in b)=>a=b. Note that some texts (e.g., Devlin ...

The axiom of Zermelo-Fraenkel set theory which asserts the existence of a set containing all the natural numbers, exists x(emptyset in x ^ forall y in x(y^' in x)), where ...