Mapping the Upper Half plane to the Unit Ball Consider the sets $\mathbb{H} = \{ \ (x_{1}, ..., x_{n}) \in \mathbb{R}^{n} \ | \ x_{n} \geq 0\ \}$ and $B_{1}(\mathbf{0}) = \{ \ \mathbf{x} \in \mathbb{R}^{n}\ | \ ||\mathbf{x}|| \leq 1 \ \}$.

Does there exist a smooth homeomorphism $\alpha : \mathbb{H} \to B_{1}(\mathbf{0})$?

I don't think that there is.
Yes, there is. Think inverse stereographic projection. (this is for the open ball, and open half-space; misread the question).
Yes, there is. Think inverse stereographic projection. (this is for the open ball, and open half-space; misread the question).