Banach space is an important term in the field of mathematics, specifically in functional analysis. It is a term used to describe a complete normed vector space. Banach spaces are used in many areas of mathematics and have applications in physics, engineering, and computer science.
Definitions
A Banach space is a complete normed vector space. This means that the space is equipped with a norm, which is a mathematical function that assigns a non-negative real number to each vector in the space. The norm satisfies certain properties, such as the triangle inequality. Completeness means that every Cauchy sequence in the space converges to a limit that is also in the space.
Origin
The term Banach space is named after Stefan Banach, a Polish mathematician who worked in functional analysis. Banach made important contributions to the theory of Banach spaces in the early 20th century.
Meaning in different dictionaries
In the Oxford English Dictionary, Banach space is defined as “a complete normed vector space, used in functional analysis.” In Merriam-Webster’s Dictionary, it is defined as “a complete normed vector space in which the distance between any two points is defined in terms of a norm and which is used especially in functional analysis.”
Associations
Banach spaces are associated with functional analysis, which is a branch of mathematics that studies spaces of functions and mappings between them. They are also associated with the study of linear operators and their properties.
Synonyms
There are no commonly used synonyms for Banach space.
Antonyms
There are no commonly used antonyms for Banach space.
The same root words
Banach space is named after Stefan Banach, a Polish mathematician. There are no other commonly used words with the same root as Banach space.
Example Sentences
- The study of Banach spaces is an important part of functional analysis.
- A Hilbert space is an example of a Banach space.
- The properties of Banach spaces are used in the study of linear operators.
- The completeness of Banach spaces makes them useful in many areas of mathematics and science.
- Banach spaces have applications in physics, engineering, and computer science.