Smooth functions into infinite dimensional Hilbert spaces

Question

I am currently working with infinite-dimensional (separable) Hilbert spaces $\mathcal{H}$ and functions of the type \begin{align} f:\mathbb{R}^n\rightarrow\mathcal{H}. \end{align} What does it mean for $f$ to be smooth? Is is sufficient to show that

1. $x\mapsto (\Phi,f(x))$ is smooth for all $\Phi\in\mathcal{H}$?
2. $x\mapsto (\Phi,f(x))$ is smooth for all $\Phi$ in some dense subset?
3. $x\mapsto \|f(x)\|$ is smooth for all $\Phi\in\mathcal{H}$?
4. $x\mapsto \|f(x)\|$ is smooth for all $\Phi$ in some dense subset?

No 1. and 2. seem plausible to me, while 3. and 4. rather not.

Is there a concept of vector-valued Schwartz functions?

Maybe someone can point me towards sources on this topic.