1.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski

2.
Hilbert space
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The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of algebra and calculus from the two-dimensional Euclidean plane. A Hilbert space is a vector space possessing the structure of an inner product that allows length. Furthermore, Hilbert spaces are complete, there are limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces, 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 —and ergodic theory, john von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis, geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space, at a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be specified by its coordinates with respect to a set of coordinate axes. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of the space of sequences that are square-summable. The latter space is often in the literature referred to as the Hilbert space. One of the most familiar examples of a Hilbert space is the Euclidean space consisting of vectors, denoted by ℝ3. The dot product takes two vectors x and y, and produces a real number x·y, If x and y are represented in Cartesian coordinates, then the dot product is defined by ⋅ = x 1 y 1 + x 2 y 2 + x 3 y 3. The dot product satisfies the properties, It is symmetric in x and y, x · y = y · x. It is linear in its first argument, · y = ax1 · y + bx2 · y for any scalars a, b, and vectors x1, x2, and y. It is positive definite, for all x, x · x ≥0, with equality if. An operation on pairs of vectors that, like the dot product, a vector space equipped with such an inner product is known as a inner product space. Every finite-dimensional inner product space is also a Hilbert space, multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist

3.
Banach space
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In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn, Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces, the vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. All norms on a vector space are equivalent. Every finite-dimensional normed space over R or C is a Banach space, if X and Y are normed spaces over the same ground field K, the set of all continuous K-linear maps T, X → Y is denoted by B. In infinite-dimensional spaces, not all maps are continuous. For Y a Banach space, the space B is a Banach space with respect to this norm, if X is a Banach space, the space B = B forms a unital Banach algebra, the multiplication operation is given by the composition of linear maps. If X and Y are normed spaces, they are isomorphic normed spaces if there exists a linear bijection T, X → Y such that T, if one of the two spaces X or Y is complete then so is the other space. Two normed spaces X and Y are isometrically isomorphic if in addition, T is an isometry, the Banach–Mazur distance d between two isomorphic but not isometric spaces X and Y gives a measure of how much the two spaces X and Y differ. Every normed space X can be embedded in a Banach space. More precisely, there is a Banach space Y and an isometric mapping T, X → Y such that T is dense in Y. If Z is another Banach space such that there is an isomorphism from X onto a dense subset of Z. This Banach space Y is the completion of the normed space X, the underlying metric space for Y is the same as the metric completion of X, with the vector space operations extended from X to Y. The completion of X is often denoted by X ^, the cartesian product X × Y of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are used, such as ∥ ∥1 = ∥ x ∥ + ∥ y ∥, ∥ ∥ ∞ = max. In this sense, the product X × Y is complete if and only if the two factors are complete. If M is a linear subspace of a normed space X, there is a natural norm on the quotient space X / M

4.
Operator topologies
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In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B of bounded linear operators on a Hilbert space H. Let be a sequence of operators on the Hilbert space H. Consider the statement that Tn converges to some operator T in H. This could have different meanings, If ∥ T n − T ∥ →0, that is. If T n x → T x for all x in H, finally, suppose T n x → T x in the weak topology of H. This means that F → F for all linear functionals F on H, in this case we say that T n → T in the weak operator topology. All of these notions make sense and are useful for a Banach space in place of the Hilbert space H, there are many topologies that can be defined on B besides the ones used above. These topologies are all convex, which implies that they are defined by a family of seminorms. In analysis, a topology is called if it has many open sets and weak if it has few open sets, so that the corresponding modes of convergence are, respectively, strong. The diagram on the right is a summary of the relations, the Banach space B has a predual B*, consisting of the trace class operators, whose dual is B. The seminorm pw for w positive in the predual is defined to be 1/2, If B is a vector space of linear maps on the vector space A, then σ is defined to be the weakest topology on A such that all elements of B are continuous. The norm topology or uniform topology or uniform operator topology is defined by the usual norm ||x|| on B and it is stronger than all the other topologies below. The weak topology is σ, in words the weakest topology such that all elements of the dual B* are continuous. It is the topology on the Banach space B. It is stronger than the ultraweak and weak operator topologies and it is stronger than the weak operator topology. The strong* operator topology or strong* topology is defined by the seminorms ||x|| and ||x*|| for h in H and it is stronger than the strong and weak operator topologies. The strong operator topology or strong topology is defined by the seminorms ||x|| for h in H and it is stronger than the weak operator topology. The weak operator topology or weak topology is defined by the seminorms || for h1, the continuous linear functionals on B for the weak, strong, and strong* topologies are the same, and are the finite linear combinations of the linear functionals for h1, h2 in H

5.
Topological vector space
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In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure with the concept of a vector space. Hilbert spaces and Banach spaces are well-known examples, unless stated otherwise, the underlying field of a topological vector space is assumed to be either the complex numbers C or the real numbers R. Some authors require the topology on X to be T1, it follows that the space is Hausdorff. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the category of topological vector spaces over a given topological field K is commonly denoted TVSK or TVectK. The objects are the vector spaces over K and the morphisms are the continuous K-linear maps from one object to another. Every normed vector space has a topological structure, the norm induces a metric. This is a vector space because, The vector addition +, V × V → V is jointly continuous with respect to this topology. This follows directly from the triangle inequality obeyed by the norm, the scalar multiplication ·, K × V → V, where K is the underlying scalar field of V, is jointly continuous. This follows from the inequality and homogeneity of the norm. Therefore, all Banach spaces and Hilbert spaces are examples of vector spaces. There are topological spaces whose topology is not induced by a norm. These are all examples of Montel spaces, an infinite-dimensional Montel space is never normable. A topological field is a vector space over each of its subfields. A cartesian product of a family of vector spaces, when endowed with the product topology, is a topological vector space. For instance, the set X of all functions f, R → R, with this topology, X becomes a topological vector space, called the space of pointwise convergence. The reason for this name is the following, if is a sequence of elements in X, then fn has limit f in X if and only if fn has limit f for every real number x. This space is complete, but not normable, indeed, every neighborhood of 0 in the topology contains lines

6.
Uniform convergence
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In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. Loosely speaking, this means that f n converges to f at a speed on its entire domain. In contrast, we say that f n converges to f pointwise, if exists a N. It is clear from these definitions that uniform convergence of f n to f on E implies pointwise convergence for every x ∈ E, frequently, no special symbol is used, and authors simply write f n → f u n i f o r m l y. The difference between the two types of convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning, Uniform convergence to a function on a given interval can be defined in terms of the uniform norm. Completely standard notions of convergence did not exist at the time, when put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit. While he thought it a fact when a series converged in this way, he did not give a formal definition. Independently, similar concepts were articulated by Philipp Ludwig von Seidel, suppose E is a set and f n, E → R are real-valued functions. This is the Cauchy criterion for uniform convergence, in another equivalent formulation, if we define a n = sup x ∈ E | f n − f |, then f n converges to f uniformly if and only if a n →0 as n → ∞. The sequence n ∈ N is said to be uniformly convergent with limit f if E is a metric space and for every x in E. It is easy to see that local uniform convergence implies pointwise convergence and it is also clear that uniform convergence implies local uniform convergence. Note that interchanging the order of there exists N and for all x in the definition above results in a statement equivalent to the convergence of the sequence. In explicit terms, in the case of convergence, N can only depend on ϵ, while in the case of pointwise convergence. It is therefore plain that uniform convergence implies pointwise convergence, the converse is not true, as the example in the section below illustrates. One may straightforwardly extend the concept to functions S → M, the most general setting is the uniform convergence of nets of functions S → X, where X is a uniform space. The above-mentioned theorem, stating that the limit of continuous functions is continuous. Uniform convergence admits a simplified definition in a hyperreal setting, thus, a sequence f n converges to f uniformly if for all x in the domain of f* and all infinite n, f n ∗ is infinitely close to f ∗. Then uniform convergence simply means convergence in the norm topology