1. ## exponential map-well defined

Can someone expain to me what does it mean that the exponential map between a Lie algebra and a Lie group is well defined (in a descriptive way please)?

I would have said that they are well defined because all of them assign to each element in the domain (Lie algebra) a unique elements in the codomain (Lie group)...is that correct?

For example
$exp: \mathfrak{so}(n)\to SO(n)$
or
$exp:\mathfrak{sl}(n,\mathbb{R})\to SL(n,\mathbb{R})$
and so on
thank you

2. Yes, it is correct: For the exponential map to be well-defined, it just needs to be a function in the good old sense.

The reason someone mentions well-definedness might be that your exponential map is defined as an infinite series. In this case, you need to assure yourself that the series is convergent, since the map would otherwise not be well-defined.

3. thank you HappyJoe Yes I have added the proff for the convergence