1.
Herman Goldstine
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Herman Heine Goldstine was a mathematician and computer scientist, who was one of the original developers of ENIAC, the first of the modern electronic digital computers. Herman Heine Goldstine was born in Chicago in 1913 to Jewish parents and he attended the University of Chicago, where he joined the Phi Beta Kappa fraternity, and graduated with a degree in Mathematics in 1933, a masters degree in 1934, and a PhD in 1936. For three years he was an assistant under Gilbert Ames Bliss, an authority on the mathematical theory of external ballistics. In 1939 Goldstine began a career at the University of Michigan, until the United States entry into World War II. In 1941 he married Adele Katz, who was an ENIAC programmer and he had a daughter and a son with Adele, who died in 1964. Two years later he married secondly Ellen Watson, Goldstine died on June 16,2004 at his home in Bryn Mawr, Pennsylvania, after a long struggle with Parkinsons disease. His death was announced by the Thomas J. Watson Research Center in Yorktown Heights, New York, as a result of the United States entering World War II, Goldstine left the University of Michigan where he was a professor in July,1942 to enlist in the Army. He was commissioned a lieutenant and worked as an ordnance mathematician calculating firing tables at the Ballistic Research Laboratory at Aberdeen Proving Ground, Maryland, the firing tables were used in battle to find the appropriate elevation and azimuth for aiming artillery, which had a range of several miles. The firing table calculations were accomplished by one hundred women operating mechanical desk calculators. Each combination of gun, round and geographical region required a set of firing tables. It took about 750 calculations to compute a trajectory and each table had about 3,000 trajectories. It took human computer at least 7 hours to calculate one trajectory, Mauchly wrote a proposal and in June 1943 he and Goldstine secured funding from the Army for the project. The ENIAC was built in 30 months with 200,000 man hours, the ENIAC was huge, measuring 30 by 60 feet and weighing 30 tons with 18,000 vacuum tubes. The device could only store 20 numbers and took days to program and it was completed in late 1945 as World War II was coming to an end. In spite of disappointment that ENIAC had not contributed to the war effort, prior even to the ENIACs completion, the Army procured a second contract from the Moore School to build a successor machine known as the EDVAC. Goldstine, Mauchly, J. Presper Eckert and Arthur Burks began to study the development of the new machine in the hopes of correcting the deficiencies of the ENIAC. Unknown to Goldstine, Neumann was then working on the Manhattan Project, the calculations needed for this project were also daunting. As a result of his conversations with Goldstine, Neumann joined the study group, Neumann intended this to be a memo to the study group, but Goldstine typed it up into a 101-page document that named Neumann as the sole author

2.
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

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.
Lp space
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In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Lebesgue spaces have applications in physics, statistics, finance, engineering, in penalized regression, L1 penalty and L2 penalty refer to penalizing either the L1 norm of a solutions vector of parameter values, or its L2 norm. Techniques which use an L1 penalty, like LASSO, encourage solutions where many parameters are zero, techniques which use an L2 penalty, like ridge regression, encourage solutions where most parameter values are small. Elastic net regularization uses a penalty term that is a combination of the L1 norm, the Fourier transform for the real line, maps Lp to Lq, where 1 ≤ p ≤2 and 1/p + 1/q =1. This is a consequence of the Riesz–Thorin interpolation theorem, and is precise with the Hausdorff–Young inequality. By contrast, if p >2, the Fourier transform does not map into Lq, Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces L2 and ℓ2 are both Hilbert spaces, in fact, by choosing a Hilbert basis, one sees that all Hilbert spaces are isometric to ℓ2, where E is a set with an appropriate cardinality. The length of a vector x = in the real vector space Rn is usually given by the Euclidean norm. The Euclidean distance between two points x and y is the length ||x − y||2 of the line between the two points. In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space, the class of p-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science. For a real number p ≥1, the p-norm or Lp-norm of x is defined by ∥ x ∥ p =1 p, of course the absolute value bars are unnecessary when p is a rational number and, in reduced form, has an even numerator. The Euclidean norm from above falls into class and is the 2-norm. The L∞-norm or maximum norm is the limit of the Lp-norms for p → ∞ and it turns out that this limit is equivalent to the following definition, ∥ x ∥ ∞ = max See L-infinity. Abstractly speaking, this means that Rn together with the p-norm is a Banach space and this Banach space is the Lp-space over Rn. The grid distance or rectilinear distance between two points is never shorter than the length of the segment between them. Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm, ∥ x ∥2 ≤ ∥ x ∥1. This fact generalizes to p-norms in that the p-norm ||x||p of any vector x does not grow with p, ||x||p+a ≤ ||x||p for any vector x and real numbers p ≥1