I try to understand why the equality/inequality you can see below holds.
Let $\mathfrak{h}$ be a separable Hilbert space and define the Fock space $\mathcal{F}:= \oplus_{N=0}^\infty\otimes^N\mathfrak{h}$. For $\varphi\in\mathfrak{h}$ let $(a(\varphi), D(\sqrt{\mathcal{N}}))$ and $(a^\dagger(\varphi), D(\sqrt{\mathcal{N}}))$ be the annihilation and creation operator with $\mathcal{N}$ the number operator. Then for all $\Psi \in D(\sqrt{\mathcal{N}}))$ is holds that
\begin{align} \vert\vert a^\dagger(\varphi)\Psi\vert\vert_\mathcal F &= \vert\vert\varphi\vert\vert_\mathfrak h \cdot \vert\vert \sqrt{\mathcal N + 1}\Psi \vert\vert_\mathcal F\\ \vert\vert a(\varphi)\Psi\vert\vert_\mathcal F &\leq \vert\vert\varphi\vert\vert_\mathfrak h \cdot \vert\vert\sqrt{\mathcal N + 1}\Psi\vert\vert_\mathcal F \end{align}
For the first i tried to use the definition of the operator \begin{align} \vert\vert a^\dagger(\varphi)\Psi\vert\vert_\mathcal F = \vert\vert (0,\varphi\cdot \psi_0, \sqrt{2}\varphi\otimes\psi_1, \sqrt{3}\varphi\otimes\psi_2,\dots)\vert\vert \end{align} but i am note sure how to proceed and how to deal with the tensor product inside the norm.