Question

Explain the local mapping and open mapping theorem.

Answer 1

Answer:

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.

Answer 2

The local mapping and Open mapping theorem are

* Open mapping

• open mapping theorem: If X and Y are Branch spaces and A: X - Y is subjective continuous linear operator , then A us an open map.

* Local mapping

• f is analytic at z0 and that

f(z)- w0 has a zero of order n at z0. Then for all sufficiently small > 0 there

exist 0 such that for all Nthe equation f(z) = w has n distinct roots.