## Separable Space Wikipedia

### Separable space - Wikipedia.

First examples. Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An important example of an uncountable separable space is the real line, in which the rational numbers form a countable dense subset. Similarly the set of all vectors = (, ...,) of which is a countable dense subset; so for every ....

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

### Totally bounded space - Wikipedia.

In metric spaces. A metric space (,) is totally bounded if and only if for every real number >, there exists a finite collection of open balls in M of radius whose union contains M.Equivalently, the metric space M is totally bounded if and only if for every >, there exists a finite cover such that the radius of each element of the cover is at most .This is equivalent to the existence of a ....

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

### Polish space - Wikipedia.

In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians--Sierpinski, Kuratowski, Tarski and others..

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

### Space (mathematics) - Wikipedia.

In ancient Greek mathematics, "space" was a geometric abstraction of the three-dimensional reality observed in everyday life. About 300 BC, Euclid gave axioms for the properties of space. Euclid built all of mathematics on these geometric foundations, going so far as to define numbers by comparing the lengths of line segments to the length of a chosen reference segment..

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

### σ-algebra - Wikipedia.

In mathematical analysis and in probability theory, a ?-algebra (also ?-field) on a set X is a nonempty collection ? of subsets of X closed under complement, countable unions, and countable intersections.The pair (X, ?) is called a measurable space.The ?-algebras are a subset of the set algebras; elements of the latter only need to be closed under the union or ....

https://en.wikipedia.org/wiki/%CE%A3-algebra.

### Nuclear space - Wikipedia.

Definition 3: A nuclear space is a topological vector space with a topology defined by a family of Hilbert seminorms, such that for any Hilbert seminorm we can find a larger Hilbert seminorm so that the natural map from to is Hilbert-Schmidt. If we are willing to use the concept of a nuclear operator from an arbitrary locally convex ....

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

### Banach space - Wikipedia.

Since every Banach space is a Frechet space, this is also true of all infinite-dimensional separable Banach spaces, including the separable Hilbert 2 sequence space with its usual norm ? ?, where (in sharp contrast to finite-dimensional spaces) () is also homeomorphic to its unit sphere {(): ? ? =}..

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

### Rumspringa - Wikipedia.

Rumspringa (Pennsylvania German pronunciation: ['r?m??prIn?]), also spelled Rumschpringe or Rumshpringa, is a rite of passage during adolescence, translated from originally Palatine German and other Southwest German dialects to English as "jumping or hopping around", used in some Amish communities. The Amish, a subsect of the Anabaptist Christian movement, intentionally ....

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

### Operator norm - Wikipedia.

The norm on the left is the one in and the norm on the right is the one in .Intuitively, the continuous operator never increases the length of any vector by more than a factor of . Thus the image of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as bounded operators. ....

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

### ER = EPR - Wikipedia.

Overview. The conjecture was proposed by Leonard Susskind and Juan Maldacena in 2013. They proposed that a wormhole (Einstein-Rosen bridge or ER bridge) is equivalent to a pair of maximally entangled black holes.EPR refers to quantum entanglement (EPR paradox).. The symbol is derived from the first letters of the surnames of authors who wrote the first paper on ....

https://en.wikipedia.org/wiki/ER_%3D_EPR.

### Moment (mathematics) - Wikipedia.

In mathematics, the moments of a function are quantitative measures related to the shape of the function's graph.If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the centre of mass, and the second moment is the moment of inertia.If the function is a probability distribution, then the first moment is the ....

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

### Quantum state - Wikipedia.

In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior..

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

### Political spectrum - Wikipedia.

A political spectrum is a system to characterize and classify different political positions in relation to one another. These positions sit upon one or more geometric axes that represent independent political dimensions. The expressions political compass and political map are used to refer to the political spectrum as well, especially to popular two-dimensional models of it..

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

### 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.

### 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.

### Real number - Wikipedia.

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion).The adjective real in this context was introduced in the 17th century by Rene Descartes, who distinguished between real and imaginary roots of polynomials. The real ....

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

### 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.

### Discrete space - Wikipedia.

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is ....

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

### Glossary of areas of mathematics - Wikipedia.

Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers. This glossary is alphabetically sorted. This hides a large part of the relationships ....

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