bounded inverse theorem
Proof : is a surjective continuous operator between the Banach spaces and . Therefore, by the open mapping theorem, takes open sets to open sets.
So, for every open set , is open in .
It is usually of great importance to know if a bounded operator has a bounded inverse. For example, suppose the equation
has unique solutions for every given . Suppose also that the above equation is very difficult to solve (numerically) for a given , but easy to solve for a value ”near” . Then, if is continuous, the correspondent solutions and are also ”near” since
Therefore we can solve the equation for a ”near” value instead, without obtaining a significant error.
|Title||bounded inverse theorem|
|Date of creation||2013-03-22 17:30:51|
|Last modified on||2013-03-22 17:30:51|
|Last modified by||asteroid (17536)|
|Synonym||inverse mapping theorem|