|
In mathematics, real coordinate space of dimensions, written R ( ) (R with superscript ''n'', also written with blackboard bold R) or R^n with keyboard is a coordinate space that allows several () real variables to be treated as a single variable. With various numbers of dimensions (sometimes unspecified), is used in many areas of pure and applied mathematics, as well as in physics. It is the prototypical real vector space and a frequently used . Due to the latter fact, geometric metaphors are widely used for , namely a plane for and three-dimensional space for . == Definition and uses == For any natural number , the set consists of all -tuples of real numbers (). It is called (the) "-dimensional real space". Depending on its construction from instances of the set , it inherits some of the latter's structure, notably: * When defined as the direct sum of vector spaces, addition and scalar multiplication are defined on : see below * is a topological space: see below An element of is written : where each is a real number. For each there exists only one , ''the'' real -space.〔Unlike many situations in mathematics where a certain object is unique up to isomorphism, is unique in the strong sense: any of its elements is described explicitly with its real coordinates.〕 Purely mathematical uses of can be roughly classified as follows, although these uses overlap. First, linear algebra studies its own properties under vector addition and linear transformations and uses it as a model of any -dimensional real vector space. Second, it is used in mathematical analysis to represent the domain of a function of real variables in a uniform way, as well as a space to which the graph of a real-valued function of real variables is a subset. The third use parametrizes geometric points with elements of ; it is common in analytic, differential and algebraic geometries. , together with supplemental structures on it, is also extensively used in mathematical physics, dynamical systems theory, mathematical statistics and probability theory. In applied mathematics, numerical analysis, and so on, arrays, sequences, and other collections of numbers in applications can be seen as the use of too. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「real coordinate space」の詳細全文を読む スポンサード リンク
|