In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein’s general theory of relativity, and is named for the Greek mathematician Euclid of Alexandria.
Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. In modern mathematics, it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. This approach brings the tools of algebra and calculus to bear on questions of geometry, and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions.
From the modern viewpoint, there is essentially only one Euclidean space of each dimension. In dimension one this is the real line; in dimension two it is the Cartesian plane; and in higher dimensions it is the real coordinate space with three or more real number coordinates. Thus a point in Euclidean space is a tuple of real numbers, and distances are defined using the Euclidean distance formula. Mathematicians often denote the n-dimensional Euclidean space by , or sometimes if they wish to emphasize its Euclidean nature. Euclidean spaces have finite dimension.
In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function.
For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the ordering matters. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,…). Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.
Baron Augustin-Louis Cauchy (21 August 1789 – 23 May 1857; French pronunciation: [oɡystɛ̃ lwi koʃi]) was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors. He defined continuity in terms of infinitesimals and gave several important theorems in complex analysis and initiated the study of permutation groups in abstract algebra. A profound mathematician, Cauchy exercised a great influence over his contemporaries and successors. His writings cover the entire range of mathematics and mathematical physics.
Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks. He was a devout Roman Catholic, strict Bourbon royalist, and a close associate of the Jesuit order.
Complete metric space
In mathematical analysis, a metric space M is called complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.
Intuitively, a space is complete if there are no “points missing” from it (inside or at the boundary). Thus, a complete metric space is analogous to a closed set. For instance, the set of rational numbers is not complete, because e.g. is “missing” from it, even though one can construct a Cauchy sequence of rational numbers that converges to it. (See the examples below.) It is always possible to “fill all the holes”, leading to the completion of a given space, as explained below.
In mathematics, a Cauchy sequence (pronounced [koˈʃi]), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.
The utility of Cauchy sequences lies in the fact that in a complete metric space (one where all such sequences are known to converge to a limit), the criterion for convergence depends only on the terms of the sequence itself. This is often exploited in algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates.
The notions above are not as unfamiliar as they might at first appear. The customary acceptance of the fact that any real number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers (whose terms are the successive truncations of the decimal expansion of x) has the real limit x. In some cases it may be difficult to describe x independently of such a limiting process involving rational numbers.
Extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ (read as positive infinity and negative infinity respectively). The projective extended real number system adds a single object, ∞ (infinity) and makes no distinction between “positive” or “negative” infinity. These new elements are not real numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted R or [−∞, +∞].
When the meaning is clear from context, the symbol +∞ is often written simply as ∞.