vol 4: Glossary
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Function
[Hazewinkel, Function] [Borowski, Function]
A function is the mathematical tool for modelling action or change. A function is a relation between two sets (or a set and itself) that associates a unique element f(x) of the second set with each element x of the first set. A function can therefore be represented as a set of ordered pairs {x, f(x)}.
Some functions can be expressed by a rule, such as f(x) = x + 1, which (when applied to the natural numbers) simply transforms each number, eg 2, into its successor, eg 3. Other functions like f(x) = sin x and so on can also be compactly represented by a rule. The existence of a rule is not a necessary property of a function, however, so that some functions need to be represented by 'look-up tables' which are simply the set of ordered pairs {x, f(x)} that define the function.
The class of functions at the foundation of this site are permutations. Some permutations can be represented succinctly by rules; others, without rules by look-up tables.
Books
| Borowski, Ephraim J, and Jonathan M Borwein, Collins Dictionary of Mathematics, Harper Collins 1089 'It is the immodest hope of the authors that this dictionary will not only prove valuable as a reference book for students of mathematics at all levels from secondary schools to a master's degree, but also offer much to interest a more general readership. Amazon back |
| Hazewinkel, Michiel, and (managing editor), Encyclopaedia of Mathematics (6 volumes), Kluwer Academic and Toppan 1995 'The Encyclopaedia of mathematics aims to be a reference work for all parts of mathematics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-85.' Amazon back |
| Kolmogorov, A N, and S V Fomin, Elements of the Theory of Functions and Functional Analysis Volumes 1 and 2, (Two volumes bound as one), Dover 1999 Jacket: Beginning with a brief introduction to set theory and mappings, the authors offer a clear presentation of the theory of metric and complete metric spaces, The principle of contraction mappings and its applications to the proof of existence theorems in the theory of differential and integral equations receives detailed analysis, as do continuous curves in metric spaces - a topic seldom discussed in textbooks. ... Part two focusses on an exposition of measure theory, the Lebesgue interval and Hilbert space. Both parts feature numerous exercises at the end of each section and include helpful lists of symbols, definitions and theorems.' Amazon back |
| Kreyszig, Erwin, Introductory Functional Analysis with Applications, John Wiley and Sons 1989 Amazon: 'Kreyszig's "Introductory Functional Analysis with Applications", provides a great introduction to topics in real and functional analysis. This book is part of the Wiley Classics Library and is extremely well written, with plenty of examples to illustrate important concepts. It can provide you with a solid base in these subjects, before one takes on the likes of Rudin and Royden. I had purchased a copy of this book, when I was taking a graduate course on real analysis and can only strongly recommend it to anyone else.' Krishnan S. Kartik Amazon back |
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