Finding inverse of linear mapping

[Somefantastik]

so for a mapping f:X->Y

where X,Y are Normed Vector Spaces

if I have a function f(x) = y such that x in X and y in Y, how do I explicitly find f inverse?

I sat down to do this and realize I've only been trained in the Reals where you switch the x,y and then solve for y. But this won't work for my function as my x,y are vectors (not scalars) and I don't have all the operations available like 1/x and ln(x).

 

[rasmhop]

By a mapping do you mean a bounded linear map? And by inverse do you require your inverse to be bounded linear as well? Maybe you are only working with isometries?

If you do not impose more conditions on f, X or Y than that f is bounded linear, then the problem is impossible in general. Just let X be the trivial normed vector space {0} and let Y be the normed vector space $\mathbb{C}$. Then we may define $f : X \to Y$ by f(0) = 0. Clearly however f cannot have an inverse because X and Y are not of equal cardinality. The only function $g : Y \to X$ is g(y) = 0, but then f(g(1)) = f(0) = 0 so g is not a right inverse.

 

[Somefantastik]

I hope my function is bounded and continuous. My ultimate goal is to show a homeomorphism from a normed vector space to another. So the function I pick must be bicontinuous (and therefore bounded). So let's assume I was smart enough to pick a bicontinuous and bounded function.

 

[rasmhop]

That is sufficient. Since f is a homeomorphism it has a continuous inverse g : Y -> X. You can show fairly easily that g is linear. g is linear and continuous and therefore bounded.

Whether this is explicit enough or not I do not know, but I doubt you will find a much more explicit construction unless you are willing to severely restrict what X and Y are allowed to be.

 

[blue_raver22]

Finding the inverse of a linear mapping in normed vector spaces can be a bit more complicated than in the real numbers. However, there are still some general steps that can be followed to find the inverse function.

Firstly, it is important to note that a linear mapping between two normed vector spaces can be represented by a matrix. So, one approach to finding the inverse function would be to find the inverse of this matrix.

Another approach would be to use the concept of the adjoint operator. The adjoint operator of a linear mapping is defined as the transpose of the matrix representing the mapping. So, by finding the adjoint operator, we can effectively find the inverse function.

Additionally, there are some specific properties that can be used to simplify the process of finding the inverse function. For example, if the linear mapping is invertible, then its inverse will also be a linear mapping. This means that we can use techniques such as Gaussian elimination to find the inverse function.

In summary, finding the inverse of a linear mapping in normed vector spaces may require some additional techniques and properties, but it is still possible to determine the inverse function. It may be helpful to consult with a mathematician or refer to additional resources for specific examples and methods.