Xah Math Notes

By Xah Lee. Date: . Last updated: .

Homotopy

Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós “same, similar” and τόπος tópos “place”) if one can be “continuously deformed” into the other, such a deformation being called a homotopy between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

quotient space topology

In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or “gluing together” certain points of a given topological space. The points to be identified are specified by an equivalence relation. This is commonly done in order to construct new spaces from given ones. The quotient topology consists of all sets with an open preimage under the canonical projection map that maps each element to its equivalence class.

Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function H : X × [0,1] → Y from the product of the space X with the unit interval [0,1] to Y such that, if x ∈ X then H(x,0) = f(x) and H(x,1) = g(x).

If we think of the second parameter of H as time then H describes a continuous deformation of f into g: at time 0 we have the function f and at time 1 we have the function g. We can also think of the second parameter as a “slider control” that allows us to smoothly transition from f to g as the slider moves from 0 to 1, and vice versa.

exceptional line

domain is the set of input of a function

range is the set of output of a function

codomain is the set the function's range lies. It is a superset of range. Basically, codamain means the “target” set of the function.

injenction (aka one-to-one function), is a function such that any output has only 1 unique input.

surjenction (aka an “onto” function), is a function such that its codomain equals to its range. That is, it covers the entire target set.

bijection (aka “one-to-one and onto” function), is both injection and surjection. That is, one-to-one and onto. Bijection pairs each element in domain and codomain.

     •   invertible •  self • differentiable
homo •
endo  • map to self
auto  • invertible and self
iso  • invertible
diffeo • isomorphism of smooth manifolds

Homomorphism

homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).

Homomorphisms of vector spaces are also called linear maps.

Being an isomorphism, an automorphism, or an endomorphism is a property of some homomorphisms, which may be defined in a way that may be generalized to any class of morphisms.

Isomorphism

isomorphism is a homomorphism that admits an inverse.

An automorphism is an isomorphism whose source and target coincide.

The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.

For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if and only if it is bijective.

Endomorphism

endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map, and an endomorphism of a group is a group homomorphism.

automorphism

An invertible endomorphism of X is called an automorphism.

Diffeomorphism

a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.

Weierstrass function

the Weierstrass function is an example of a pathological real-valued function on the real line. The function has the property of being continuous everywhere but differentiable nowhere. It is named after its discoverer Karl Weierstrass.

Historically, the Weierstrass function is important because it was the first published example (1872) to challenge the notion that every continuous function was differentiable except on a set of isolated points.[1]

Plateau problem

Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem is considered part of the calculus of variations. The existence and regularity problems are part of geometric measure theory.

Open mapping theorem (functional analysis)

In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map. More precisely, (Rudin 1973, Theorem 2.11):

Open Mapping Theorem. If X and Y are Banach spaces and A : X → Y is a surjective continuous linear operator, then A is an open map (i.e. if U is an open set in X, then A(U) is open in Y).

Bounded operator

Bounded operator

In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X. In other words, there exists some M ≻ 0 such that for all v in X

{\displaystyle \|Lv\|_{Y}\leq M\|v\|_{X}.\,\,} \|Lv\|_Y \le M \|v\|_X.\, \, The smallest such M is called the operator norm {\displaystyle \|L\|_{\mathrm {op} }\,} \|L\|_{\mathrm{op}} \, of L.

A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v)=0 for all v. Rather, a bounded linear operator is a locally bounded function.

A linear operator between normed spaces is bounded if and only if it is continuous, and by linearity, if and only if it is continuous at zero.

Banach space

A Banach space is a vector space X over the field R of real numbers, or over the field C of complex numbers, which is equipped with a norm and which is complete with respect to that norm, that is to say, for every Cauchy sequence {xn} in X, there exists a element x in X such that

{\displaystyle \lim _{n\to \infty }x_{n}=x,} \lim _{n\to \infty }x_{n}=x,

or equivalently:

{\displaystyle \lim _{n\to \infty }\left\|x_{n}-x\right\|_{X}=0.} \lim _{n\to \infty }\left\|x_{n}-x\right\|_{X}=0.

Hilbert Space

Riemann sphere

the Riemann sphere, named after Bernhard Riemann,[1] is a model of the extended complex plane, the complex plane plus a point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point “∞” is near to very large numbers, just as the point “0” is near to very small numbers.