Mathematical Structures - Vector Space, Group, Metric Space, Topology, Ring, Fractal, Monoid, Measure, Algebraic Structure, Lattice, Building (Paperback)

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 69. Chapters: Vector space, Group, Metric space, Topology, Ring, Fractal, Monoid, Measure, Algebraic structure, Lattice, Building, Module, Biordered set, Periodic matrix set, Mathematical structure, Weakly o-minimal structure, Natural topology, Cut locus, Prosolvable group, Ideal ring bundle, Invertible module, Argand system, Higraph, Bernstein set. Excerpt: A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex numbers, rational numbers, or even more general fields instead. The operations of vector addition and scalar multiplication have to satisfy certain requirements, called axioms, listed below. An example of a vector space is that of Euclidean vectors which are often used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real factor is another force vector. In the same vein, but in more geometric parlance, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vector spaces are the subject of linear algebra and are well understood from this point of view, since vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. The theory is further enhanced by introducing on a vector space some additional structure, such as a norm or inner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide if a sequence of...