For $n\geq 2$, we let $\mathcal{H}$ be the complex vector space of all complex-valued functions on $[0,1]$ such that (a) $f(0)=0$, (b) for $1\leq k\leq n-1$, $f^{(k)}$ exists everywhere and is continuous, and (c) $f^{(n-1)}$ is absolutely continuous and (the a.e.-defined) $f^{(n)}$ is in $L^2$. We define an inner product on $\mathcal{H}$ by $$ \langle f,g\rangle=\sum_{k=1}^n\int_0^1f^{(k)}(x)\overline{g^{(k)}(x)}dx, $$ and let $||\cdot||$ be the induced norm.
Then $\mathcal{H}$ is a vector space and the inner product is well-defined. To show that the normed space is complete, I take a series $\sum_j f_j$ such that $\sum_j ||f_j||<\infty$, and show that it converges to a function in $\mathcal{H}$.
I've shown this provided that $\sum_j ||f_j^{(k)}||_2<\infty$ for $1\leq k\leq n$. But is $\sum_j ||f_j^{(k)}||_2<\infty$ actually implied by $\sum_j ||f_j||<\infty$? (Here $L^2\subset L^1$.)
Hint: $$\|f\|^2 = \sum_{k=1}^n \|f^{(k)}\|_2^2 \ge \|f^{(k)}\|_2^2$$