Banach Space
Definition and Overview
A Banach space is a complete normed vector space. This concept is named after the Polish mathematician Stefan Banach who introduced it in the early 20th century. In simpler terms, a Banach space is a playground in which mathematicians and physicists can perform calculations involving vectors and scalars. It is a set of elements, such as vectors, along with operations of vector addition and scalar multiplication, which satisfy certain axioms. In addition, every Cauchy sequence in this space converges to an element of the space.
Properties
A Banach space is defined by three properties: it is a vector space, it has a norm, and it is complete.
Vector Space
A vector space is a set of elements, called vectors, which can be added together and multiplied by scalars (real or complex numbers), and these operations satisfy eight axioms. These axioms include properties like associativity of addition, commutativity of addition, identity element of addition, and distributivity of scalar multiplication with respect to vector addition.
Norm
A norm on a vector space is a function that assigns a strictly positive length or size to each vector in the vector space - except for the zero vector which is assigned a length of zero. A norm must satisfy three properties: it is positive definite, homogeneous and it obeys the triangle inequality.
Completeness
A normed vector space is said to be complete, or a Banach space, if every Cauchy sequence of points in that space converges to a limit that is also in that space. In other words, all sequences in the space that should have a limit, do have a limit in the space.
Examples
There are many examples of Banach spaces in mathematics. Some of the most common ones include:
- The real numbers, R, with the standard absolute value, are a Banach space. - The complex numbers, C, with the absolute value, are a Banach space. - For any positive integer p, the space Lp(μ) of measurable functions f for which the pth power of the absolute value is integrable is a Banach space. This is known as Lp spaces. - The space of continuous functions on a closed and bounded interval [a, b] is a Banach space when equipped with the supremum norm.
Banach's Fixed Point Theorem
One of the most important results in Banach spaces is the Banach's Fixed Point Theorem, also known as the contraction mapping principle. This theorem provides a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.
Applications
Banach spaces are used extensively in functional analysis, and have applications in many other areas of mathematics such as topology, abstract algebra, stochastic analysis, and measure theory. They are also used in quantum mechanics where states of a quantum system are represented by vectors in a Banach space, and observables are represented by operators on these spaces.