Banach space of continuously differentiable function on $\Omega \subset \mathbb{R}^n$

Question

I am a bit confused how the Banach space of continuously differentiable functions on $\Omega \subset R^n$ (possible unbounded), $n\geq 2$ is defined.

Let us assume that $n=2$ and $\Omega=R^2.$ Is it too much to demand that $f(x,y)$ is continuous and its partial derivatives are continuous functions on $\Omega$, with

$$ \| f\| =\sup \limits_{(x,y)\in \Omega} \left(|f(x,y)|+ |\frac{\partial f}{\partial x}(x,y)|+|\frac{\partial f}{\partial y}(x,y)|\right) $$

Would appreciate some help here... Thank you!

Answer 1

Is the classic definition. See for example Banach Spaces III: Banach Spaces of Continuous Functions. A norm is by definition real-valued, so if $\|\cdot\| = \sup(\cdots) = \infty$, it is not a norm.