## Banach Space Wikipedia

### Banach space - Wikipedia.

Definition. A Banach space is a complete normed space (, ? ?). A normed space is a pair (, ? ?) consisting of a vector space over a scalar field (where is commonly or ) together with a distinguished norm ? ?:. Like all norms, this norm induces a translation invariant distance function, called the canonical or induced metric, defined by.

https://en.wikipedia.org/wiki/Banach_space.

### Lp space - Wikipedia.

Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus.The spaces L 2 and l 2 are both Hilbert spaces. In fact, by choosing a Hilbert basis E, i.e., a maximal orthonormal subset of L 2 or any Hilbert space, one sees that every Hilbert space is isometrically isomorphic to l 2 (E) (same E as above), i.e., a Hilbert space of type l 2..

https://en.wikipedia.org/wiki/Lp_space.

### Stefan Banach - Wikipedia.

Stefan Banach (Polish: ['stefan 'banax] (); 30 March 1892 - 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original member of the Lwow School of Mathematics.His major work was the 1932 book, Theorie des ....

https://en.wikipedia.org/wiki/Stefan_Banach.

### Nuclear space - Wikipedia.

If the completion in this norm is , then there is a natural map from whenever , and this is nuclear whenever > + essentially because the series is then absolutely convergent. In particular for each norm ? ? this is possible to find another norm, say ? ? +, such that the map + is nuclear. So the space is nuclear. The space of smooth functions on any compact manifold is nuclear..

https://en.wikipedia.org/wiki/Nuclear_space.

### Banach–Tarski paradox - Wikipedia.

The Banach-Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.Indeed, the reassembly process involves only moving the ....

https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox.

### Space (mathematics) - Wikipedia.

A Banach space is a complete normed space. Many spaces of sequences or functions are infinite-dimensional Banach spaces. The set of all vectors of norm less than one is called the unit ball of a normed space. It is a convex, centrally symmetric set, generally not an ellipsoid; for example, it may be a polygon (in the plane) or, more generally ....

https://en.wikipedia.org/wiki/Space_(mathematics).

### Totally bounded space - Wikipedia.

Any topological vector space is an abelian topological group under addition, so the above conditions apply. Historically, definition 1(b) was the first reformulation of total boundedness for topological vector spaces; it dates to a 1935 paper of John von Neumann.. This definition has the appealing property that, in a locally convex space endowed with the weak topology, the ....

https://en.wikipedia.org/wiki/Totally_bounded_space.

### Simply connected space - Wikipedia.

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space ....

https://en.wikipedia.org/wiki/Simply_connected_space.

### Implicit function theorem - Wikipedia.

where is the matrix of partial derivatives in the variables and is the matrix of partial derivatives in the variables .The implicit function theorem says that if is an invertible matrix, then there are , , and as desired. Writing all the hypotheses together gives the following statement. Statement of the theorem. Let : + be a continuously differentiable function, and let + have coordinates (,)..

https://en.wikipedia.org/wiki/Implicit_function_theorem.

### Schauder fixed-point theorem - Wikipedia.

History. The theorem was conjectured and proven for special cases, such as Banach spaces, by Juliusz Schauder in 1930. His conjecture for the general case was published in the Scottish book.In 1934, Tychonoff proved the theorem for the case when K is a compact convex subset of a locally convex space. This version is known as the Schauder-Tychonoff fixed-point theorem..

https://en.wikipedia.org/wiki/Schauder_fixed-point_theorem.

### Riesz representation theorem - Wikipedia.

The Riesz representation theorem, sometimes called the Riesz-Frechet representation theorem after Frigyes Riesz and Maurice Rene Frechet, establishes an important connection between a Hilbert space and its continuous dual space.If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are ....

https://en.wikipedia.org/wiki/Riesz_representation_theorem.

### Metric (mathematics) - Wikipedia.

In mathematics, a metric or distance function is a function that gives a distance between each pair of point elements of a set.A set with a metric is a metric space. A metric induces a topology on a set, but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is a metrizable space.. One important source of metrics in differential ....

https://en.wikipedia.org/wiki/Metric_(mathematics).

### Closed graph theorem - Wikipedia.

It is said that the graph of is closed if is a closed subset of (with the product topology).. Any continuous function into a Hausdorff space has a closed graph.. Any linear map, :, between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a) is sequentially continuous in the sense of the product ....

https://en.wikipedia.org/wiki/Closed_graph_theorem.

### Espace de Banach — Wikipédia.

Comme tout espace metrique complet, un espace de Banach verifie la propriete suivante : Soit une suite decroissante de fermes non vides dont la suite des diametres tend vers 0. Alors l'intersection des fermes est non vide et reduite a un singleton.. Cette propriete permet de demontrer que tout espace metrique complet (en particulier tout espace de Banach) est de ....

https://fr.wikipedia.org/wiki/Espace_de_Banach.

### Sobolev space - Wikipedia.

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of L p-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space.Intuitively, a Sobolev space is a space of functions possessing sufficiently many ....

https://en.wikipedia.org/wiki/Sobolev_space.

### Klein bottle - Wikipedia.

In topology, a branch of mathematics, the Klein bottle (/ ' k l aI n /) is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside ....

https://en.wikipedia.org/wiki/Klein_bottle.

### Euclidean geometry - Wikipedia.

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates), and deducing many other propositions from these.Although many of Euclid's results had been stated earlier, Euclid was ....

https://en.wikipedia.org/wiki/Euclidean_geometry.

### Fréchet space - Wikipedia.

Comparison to Banach spaces. In contrast to Banach spaces, the complete translation-invariant metric need not arise from a norm.The topology of a Frechet space does, however, arise from both a total paranorm and an F-norm (the F stands for Frechet).. Even though the topological structure of Frechet spaces is more complicated than that of Banach spaces due to the ....

https://en.wikipedia.org/wiki/Fr%C3%A9chet_space.

### Cauchy–Schwarz inequality - Wikipedia.

Cauchy-Schwarz inequality [written using only the inner product]) where ? , ? {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product . Examples of inner products include the real and complex dot product ; see the examples in inner product . Every inner product gives rise to a norm , called the canonical or induced norm , where the norm of a vector u {\displaystyle \mathbf {u ....

https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality.

### Bounded operator - Wikipedia.

In topological vector spaces. A linear operator : between two topological vector spaces (TVSs) is called a bounded linear operator or just bounded if whenever is bounded in then () is bounded in . A subset of a TVS is called bounded (or more precisely, von Neumann bounded) if every neighborhood of the origin absorbs it. In a normed space (and even in a seminormed space), ....

https://en.wikipedia.org/wiki/Bounded_operator.

### Fatou's lemma - Wikipedia.

Fatou's lemma remains true if its assumptions hold -almost everywhere.In other words, it is enough that there is a null set such that the values {()} are non-negative for every . To see this, note that the integrals appearing in Fatou's lemma are unchanged if we change each function on .. Proof. Fatou's lemma does not require the monotone convergence theorem, but the latter ....

https://en.wikipedia.org/wiki/Fatou%27s_lemma.

### Jensen's inequality - Wikipedia.

In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Holder in 1889. Given its generality, the inequality appears in ....

https://en.wikipedia.org/wiki/Jensen%27s_inequality.