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Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

2.
Vector space
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A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers. The operations of addition and scalar multiplication must satisfy certain requirements, called axioms. Euclidean vectors are an example of a vector space and they represent physical quantities such as forces, any two forces can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces and these vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are commonly used. This is particularly the case of Banach spaces and Hilbert spaces, historically, the first ideas leading to vector spaces can be traced back as far as the 17th centurys analytic geometry, matrices, systems of linear equations, and Euclidean vectors. Today, vector spaces are applied throughout mathematics, science and engineering, furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques, Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra. The concept of space will first be explained by describing two particular examples, The first example of a vector space consists of arrows in a fixed plane. This is used in physics to describe forces or velocities, given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w, when a is negative, av is defined as the arrow pointing in the opposite direction, instead. Such a pair is written as, the sum of two such pairs and multiplication of a pair with a number is defined as follows, + = and a =. The first example above reduces to one if the arrows are represented by the pair of Cartesian coordinates of their end points. A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below, elements of V are commonly called vectors. Elements of F are commonly called scalars, the second operation, called scalar multiplication takes any scalar a and any vector v and gives another vector av. In this article, vectors are represented in boldface to distinguish them from scalars

3.
Dual space
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In mathematics, any vector space V has a corresponding dual vector space consisting of all linear functionals on V together with a naturally induced linear structure. The dual space as defined above is defined for all vector spaces, when defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in many branches of mathematics that use vector spaces, when applied to vector spaces of functions, dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the space is an important concept in functional analysis. Given any vector space V over a field F, the dual space V∗ is defined as the set of all linear maps φ, V → F, since linear maps are vector space homomorphisms, the dual space is also sometimes denoted by Hom. The dual space V∗ itself becomes a space over F when equipped with an addition and scalar multiplication satisfying, = φ + ψ = a for all φ and ψ ∈ V∗, x ∈ V. Elements of the dual space V∗ are sometimes called covectors or one-forms. The pairing of a functional φ in the dual space V∗ and this pairing defines a nondegenerate bilinear mapping ⟨·, ·⟩, V∗ × V → F called the natural pairing. If V is finite-dimensional, then V∗ has the dimension as V. Given a basis in V, it is possible to construct a basis in V∗. This dual basis is a set of linear functionals on V, defined by the relation e i = c i, i =1, …, n for any choice of coefficients ci ∈ F. In particular, letting in turn one of those coefficients be equal to one. For example, if V is R2, and its basis chosen to be, then e1 and e2 are one-forms such that e1 =1, e1 =0, e2 =0, and e2 =1. In particular, if we interpret Rn as the space of columns of n real numbers, such a row acts on Rn as a linear functional by ordinary matrix multiplication. One way to see this is that a functional maps every n-vector x into a number y. So an element of V∗ can be thought of as a particular family of parallel lines covering the plane. To compute the value of a functional on a given vector, or, informally, one counts how many lines the vector crosses. The dimension of R∞ is countably infinite, whereas RN does not have a countable basis, again the sum is finite because fα is nonzero for only finitely many α

4.
Absolutely convex set
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A set C in a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced, in which case it is called a disk. The absolutely convex hull of the set A is defined to be absconv A =, Vector, for vectors in physics Vector field Robertson, A. P. W. J. Robertson

5.
Absolutely convex envelope
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A set C in a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced, in which case it is called a disk. The absolutely convex hull of the set A is defined to be absconv A =, Vector, for vectors in physics Vector field Robertson, A. P. W. J. Robertson