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Bessel's Inequality
In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828. Let H be a Hilbert space, and suppose that e_1, e_2, ... is an orthonormal sequence in H. Then, for any x in H one has :\sum_^\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2, where ⟨·,·⟩ denotes the inner product in the Hilbert space H. If we define the infinite sum :x' = \sum_^\left\langle x,e_k\right\rangle e_k, consisting of "infinite sum" of vector resolute x in direction e_k, Bessel's inequality tells us that this series converges. One can think of it that there exists x' \in H that can be described in terms of potential basis e_1, e_2, \dots. For a complete orthonormal sequence (that is, for an orthonormal sequence that is a basis), we have Parseval's identity, which replaces the inequality with an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis. Intuitive overview The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be ''perpendicular'' if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero. Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces. In Cartesian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frederic Bessel
Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist, and geodesist. He was the first astronomer who determined reliable values for the distance from the sun to another star by the method of parallax. A special type of mathematical functions were named Bessel functions after Bessel's death, though they had originally been discovered by Daniel Bernoulli and then generalised by Bessel. Life and family Bessel was born in Minden, Westphalia, then capital of the Prussian administrative region Minden-Ravensberg, as second son of a civil servant into a large family. At the age of 14 Bessel was apprenticed to the import-export concern Kulenkamp at Bremen. The business's reliance on cargo ships led him to turn his mathematical skills to problems in navigation. This in turn led to an interest in astronomy as a way of determining longitude. Bessel came to the attention of a major figure of German astronomy at the time, Heinrich Wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Product Space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product naturally induces an associated norm, (denoted , x, and , y, in the picture); so, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Resolute
The vector projection of a vector on (or onto) a nonzero vector , sometimes denoted \operatorname_\mathbf \mathbf (also known as the vector component or vector resolution of in the direction of ), is the orthogonal projection of onto a straight line parallel to . It is a vector parallel to , defined as: \mathbf_1 = a_1\mathbf where a_1 is a scalar, called the scalar projection of onto , and is the unit vector in the direction of . In turn, the scalar projection is defined as: a_1 = \left\, \mathbf\right\, \cos\theta = \mathbf\cdot\mathbf where the operator ⋅ denotes a dot product, ‖a‖ is the length of , and ''θ'' is the angle between and . Which finally gives: \mathbf_1 = \left(\mathbf \cdot \mathbf\right) \mathbf = \frac \frac = \frac = \frac ~ . The scalar projection is equal to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of . The vector component or vector resolute of perpendi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality (mathematics)
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities: * The notation ''a'' ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equivalence is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or not greater than ''b''). * The notation ''a'' ≥ ''b'' or ''a'' ⩾ ''b'' means that ''a'' is greater than or equal to ''b'' (or, equivalently, at least ''b'', or not less than ''b''). The re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series (mathematics)
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Of A Sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the \lim symbol (e.g., \lim_a_n).Courant (1961), p. 29. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests. Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers. History The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes. Leucippus, Democritus, Antiphon, Eudoxus, and Archimedes developed the method of exhaustion, which uses an infinite sequence of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthonormal Basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space \R^n is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for \R^n arises in this fashion. For a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of \R^n under dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
