I am facing a problem as the following: Suppose $\mathcal{H}$ is a Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and its induced norm $|\cdot|$. Consider the space of functions $(x,t)\in\mathcal{H}\times[0,T]\mapsto\lambda(x,t)\in\mathcal{H}$ equipped with the norm \begin{align*} \|\lambda\| := \sup_{x\in\mathcal{H},t\in[0,T]}\frac{|\lambda(x,t)|}{|x|}, \end{align*} this is a Banach space. However, consider the set of function such that \begin{align*} \|\lambda\|\leq A_1,\,\,\,\|D_x\lambda(x,t)\|_{\text{linear operator}}\leq A_2, \end{align*} where $D_x$ is in Frechet sense, and the norm corresponding is the operator norm for linear operators. Is this set closed?
Thank you very much!