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Analysis is a branch of Mathematics. |
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Analysis is that branch of mathematics which deals with the real numbers and complex numbers and their functions. It has its beginnings in the rigorous formulation of calculus and studies concepts such as continuity, integration and differentiability in general settings. |
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The beginning student will best understand the difference between analysis and abstract algebra with the following description. Algebra studies structure, while analysis studies continuity. Initially, students are taught how to [mathematical proof]? simple theorems. Pythagoras was most distraught when one of his disciples showed him that the diagonal of a square of side 1 had a length that could not be expressed as the quotient of two natural numbers. This is one of the first proofs that any student of mathematics will see. |
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Initially, students are taught how to prove? simple theorems. Pythagoras was most distraught when one of his disciples showed him that the diagonal of a square of side 1 had a length that could not be expressed as the quotient of two natural numbers. This is one of the first proofs that any student of mathematics will see. |
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Notions of point set topology and metric spaces such as compactness, completeness?, connectedness, [uniform continuity]?, separability, Lipschitz maps, [contractive maps]? and so on are also investigated. Armed with these tools, it is possible to approach more abstract analysis. |
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Notions of point set topology and metric spaces such as compactness, completeness?, connectedness, [uniform continuity]?, separability, Lipschitz maps, [contractive maps]? and so on are also investigated. Armed with these tools, it is possible to approach more abstract analysis. |
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Functional analysis deals with linear operations. If U,V are normed vector spaces, then we can try to look for linear maps from U to V that are also continuous. If V is the field of scalars (either the real numbers or the complex numbers) then such a linear map is called a functional?. If U=V then such a linear map is called an operator. We usually require some more structure of U and V, perhaps that they be [Banach spaces]? or [Hilbert spaces]?. The space of all continuous linear maps from U to V is denoted by L(U,V). The space of functionals is denoted by U*. The space of all continuous operators is denoted L(U). Here we list some important results of Functional analysis. If U is a Hilbert space, then U*=U ([Riesz representation theorem]?). For Banach spaces, U** contains U and U***=U*. The [uniform boundedness principle]? is a result on sets of operators with tight bounds. The [spectral theorem]? gives an integral formula for [normal operators]? on a Hilbert space. The Hahn-Banach theorem is about extending functionals from U to V when U is a subspace of V, in a norm-preserving fashion. One of the triumphs of functional analysis was to show that the Hydrogen atom was stable. Also of interest: the [[L^p spaces]] and [Hardy spaces]? or H^p spaces. |
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Functional analysis deals with linear operations. If U,V are normed vector spaces, then we can try to look for linear maps from U to V that are also continuous. If V is the field of scalars (either the real numbers or the complex numbers) then such a linear map is called a functional?. If U=V then such a linear map is called an operator. We usually require some more structure of U and V, perhaps that they be Banach spaces or Hilbert spaces. The space of all continuous linear maps from U to V is denoted by L(U,V). The space of functionals is denoted by U*. The space of all continuous operators is denoted L(U). Here we list some important results of functional analysis: If U is a Hilbert space, then U*=U ([Riesz representation theorem]?). For Banach spaces, U** contains U and U***=U*. The [uniform boundedness principle]? is a result on sets of operators with tight bounds. The [spectral theorem]? gives an integral formula for [normal operators]? on a Hilbert space. The Hahn-Banach theorem is about extending functionals from U to V when U is a subspace of V, in a norm-preserving fashion. One of the triumphs of functional analysis was to show that the hydrogen atom was stable. Also of interest: the [[Lp spaces]] and [Hardy spaces]? or Hp spaces. |
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harmonic analysis deals with [Fourier series]?. [Jean Baptiste Joseph Fourier]?, in his work on the [heat equation]?, argued that any function could be written as a sum of sines and cosines. From his proposition, it can be argued that most of modern mathematics originated. A modern but simple introductions is as follows. If H is a Hilbert space, then a set {e_k} in H is said to be a basis if: |
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Harmonic analysis deals with [Fourier series]?. [Jean Baptiste Joseph Fourier]?, in his work on the [heat equation]?, argued that any function could be written as a sum of sines and cosines. From his proposition, it can be argued that most of modern mathematics originated. A modern but simple introductions is as follows. If H is a Hilbert space, then a set {ek} in H is said to be a basis if: |
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1) <e_j,e_k> = 0 if j \neq k and 1 if j \eq k 2) the [linear span]? of {e_k} is dense in H |
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1) <ej,ek> = 0 if j ≠ k and 1 if j = k 2) the [linear span]? of {ek} is dense in H |
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v=\sum_k <v,e_k>e_k |
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v=∑k <v,ek>ek |
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This expression on the right is called the Fourier series of v. This reduces to Fourier's version, by taking H to be a suitable space of functions, and e_k to be a suitable set of trigonometric functions. There are also other generalizations -- it turns out that there is a reconstruction formula of sorts for certain L^p spaces. In addition, if the domain is not the interval, but perhaps some strange and interesting group?, a form of Fourier decomposition is possible with basis functions chosen from the group structure of the domain. See the [Peter-Weyl theorem]?, [representation theory]?, lie groups and lie algebras. |
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This expression on the right is called the Fourier series of v. This reduces to Fourier's version, by taking H to be a suitable space of functions, and ek to be a suitable set of trigonometric functions. There are also other generalizations -- it turns out that there is a reconstruction formula of sorts for certain Lp spaces. In addition, if the domain is not the interval, but perhaps some strange and interesting group?, a form of Fourier decomposition is possible with basis functions chosen from the group structure of the domain. See the [Peter-Weyl theorem]?, [representation theory]?, lie groups and lie algebras. |
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Analysis also means, in philosophy, an account of the meaning or content of a word, phrase, or concept. In practice, analyses are not easily distinguishable from definitions. It is held by many contemporary philosophers that analyses, per se, are not possible; other terms used for the same sort of item are explication and account. See [philosophical analysis]?. |
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Analysis also means, in philosophy, an account of the meaning or content of a word, phrase, or concept. In practice, analyses are not easily distinguishable from definitions. It is held by many contemporary philosophers that analyses, per se, are not possible; other terms used for the same sort of item are explication and account. See [philosophical analysis]?. |
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