Norm of dual space of $H_0^1$

Question

Let $H^{-1}$ denote the dual space of $H_0^1(\Omega)$. Then every $f \in H^{-1}$ can be represented as $$f(u) = \int_\Omega f^0u\ dx + \sum_{k=1}^n \int_\Omega f^k \partial_k u\ dx$$ for some functions $f^0, \cdots, f^n \in L^2(\Omega)$. Now I want to show that the norm of $f \in H^{-1}(\Omega)$ can be representad as $$ \Vert f \Vert_{H_0^{-1}} = \inf \left\{ \left( \sum_{k = 0}^n \Vert f^k\Vert_{L^2}^2 \right)^{\frac{1}{2}} \colon f^0, \cdots, f^n \text{ satisfy the above representation} \right\}. $$ I have problems showing $\Vert f \Vert_{H^{-1}} \leq \inf \{\cdots\}$. I am only able to show for $u \in H_0^1$ with $\Vert u \Vert = 1$ that $$\vert f(u) \vert \leq \sum_{k = 0}^n \Vert f^k\Vert_{L^2}$$ for $f^0, \cdots f^k$ satisfying the above representation, which is not excactly what we want here. The other direction is easy as this is just applying the Riesz representation theorem. Any help is appreciated.

Answer 1

It's just applieng cauchy schwarz for finite dimensional vector spaces. $$\vert f(u) \vert \leq \int \vert f^0u \vert\ dx + \sum_{k = 1}^n\int \vert f^k \partial _ku \vert\ dx \leq \sum_{k = 0}^n \Vert f^k \Vert \cdot \Vert \partial_k u \Vert_{L^2} \leq \left( \sum_{k = 0}^{n} \Vert f^k\Vert^2 \right)^{\frac{1}{2}} \Vert u \Vert_{H^1}.$$