Let $k \in \mathbb{N}$, $p \in [1,\infty)$ and $\Omega \subset \mathbb{R}^n$ be a bounded region. Consider the Sobolev space $W^{k,p}(\Omega)$.
Then, by definition, any $f \in W^{k,p}(\Omega)$ has all weak derivatives up to the order $k$ and they are all in $L^p(\Omega)$. It is a Banach space with the norm \begin{equation} \lVert f \rVert_{k,p}:= \Bigl( \sum_{ \lvert i \rvert \leq k} \lVert f^{(i)} \rVert_{p}^p \Bigr)^{1/p} \end{equation}
However, I wonder if the following norm is equivalent to above $\lVert f \rVert_{k,p}$: \begin{equation} \lVert f \rVert'_{k,p}:=\sum_{ \lvert i \rvert \leq k} \lVert f^{(i)} \rVert_{p} \end{equation}
At least $\lVert f \rVert'_{k,p} < \infty$ is equivalent to $f \in W^{k,p}(\Omega)$. However, I do not see how $\lVert f \rVert'_{k,p}$ and $\lVert f \rVert_{k,p}$ are equivalent on $W^{k,p}(\Omega)$.
Could anyone help me?