Helly space
Encyclopedia
In mathematics, and particularly functional analysis
, the Helly space, named after Eduard Helly
, consists of all monotonically increasing function
s , where [0,1] denotes the closed interval given by the set of all x such that In other words, for all we have and also if then
Let the closed interval [0,1] be denoted simply by I. We can form the space II by taking the uncountable Cartesian product
of closed intervals:
The space II is exactly the space of functions . For each point x in [0,1] we assign the point ƒ(x) in
. The Helly space has a topology; namely the induced topology
as a subset of II.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, the Helly space, named after Eduard Helly
Eduard Helly
Eduard Helly was a mathematician and the eponym of Helly's theorem, Helly families, Helly's selection theorem, Helly metric, and the Helly–Bray theorem. In 1912, Helly published a proof of Hahn–Banach theorem, 15 years before Hahn and Banach discovered it independently...
, consists of all monotonically increasing function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s , where [0,1] denotes the closed interval given by the set of all x such that In other words, for all we have and also if then
Let the closed interval [0,1] be denoted simply by I. We can form the space II by taking the uncountable Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of closed intervals:
The space II is exactly the space of functions . For each point x in [0,1] we assign the point ƒ(x) in
Topology
The Helly space is a subset of II. The space II has its own topology, namely the product topologyProduct topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
. The Helly space has a topology; namely the induced topology
Induced topology
In topology and related areas of mathematics, an induced topology on a topological space is a topology which is "optimal" for some function from/to this topological space.- Definition :Let X_0, X_1 be sets, f:X_0\to X_1....
as a subset of II.