Consider real variables $x, y$ and a function $f(x, y) \in H^s(\mathbb{R}^2)$, say for some $s \in (0, 1)$. I am trying to get an understanding of mixed Sobolev spaces of the form $H^s_x(H^s_y)$, where the mixed Sobolev norm is defined by $$ \Vert f\Vert_{H^s_x(H^s_y)} = \Vert \Vert f\Vert_{H^s_y}\Vert_{H^s_x}.$$ In other words, $H^s_x$ functions taking values in $H^s_y$. My question is, does $f \in H^s(\mathbb{R}^2)$ automatically imply $f \in H^s_x(H^s_y)$?

It is obvious that $L^2_x(L^2_y) = L^2(\mathbb{R}^2)$, but in the case of Sobolev spaces, it is not so clear. Understanding this (relatively simple ?) case will help me understand the general theory better.

Consider real variables $x, y$ and a function $f(x, y) \in H^s(\mathbb{R}^2)$, say for some $s \in (0, 1)$. I am trying to get an understanding of mixed Sobolev spaces of the form $H^s_x(H^s_y)$, where the mixed Sobolev norm is defined by $$ \Vert f\Vert_{H^s_x(H^s_y)} = \Vert \Vert f\Vert_{H^s_y}\Vert_{H^s_x}.$$ In other words, $H^s_x$ functions taking values in $H^s_y$. My question is, does $f \in H^s(\mathbb{R}^2)$ automatically imply $f \in H^s_x(H^s_y)$?

It is obvious that $L^2_x(L^2_y) = L^2(\mathbb{R}^2)$, but in the case of Sobolev spaces, it is not so clear. Understanding this (relatively simple ?) case will help me understand the general theory better.

Edit: From Lions and Magenes, and also from Michael Renardy's answer below, I think that the correct thing to try to infer is $f \in H^s(\mathbb{R}^2) \Rightarrow f \in H^s_x(L^2_y)$. But it is not clear to me how to prove this.

Consider real variables $x, y$ and a function $f(x, y) \in H^s(\mathbb{R}^2)$, say for some $s \in (0, 1)$. I am trying to get an understanding of mixed Sobolev spaces of the form $H^s_x(H^s_y)$, where the mixed Sobolev norm is defined by $$ \Vert f\Vert_{H^s_x(H^s_y)} = \Vert \Vert f\Vert_{H^s_y}\Vert_{H^s_x}.$$ In other words, $H^s_y$ functions taking values in $H^s_x$. My question is, does $f \in H^s(\mathbb{R}^2)$ automatically imply $f \in H^s_x(H^s_y)$?

It is obvious that $L^2_x(L^2_y) = L^2(\mathbb{R}^2)$, but in the case of Sobolev spaces, it is not so clear. Understanding this (relatively simple ?) case will help me understand the general theory better.

Consider real variables $x, y$ and a function $f(x, y) \in H^s(\mathbb{R}^2)$, say for some $s \in (0, 1)$. I am trying to get an understanding of mixed Sobolev spaces of the form $H^s_x(H^s_y)$, where the mixed Sobolev norm is defined by $$ \Vert f\Vert_{H^s_x(H^s_y)} = \Vert \Vert f\Vert_{H^s_y}\Vert_{H^s_x}.$$ In other words, $H^s_x$ functions taking values in $H^s_y$. My question is, does $f \in H^s(\mathbb{R}^2)$ automatically imply $f \in H^s_x(H^s_y)$?

It is obvious that $L^2_x(L^2_y) = L^2(\mathbb{R}^2)$, but in the case of Sobolev spaces, it is not so clear. Understanding this (relatively simple ?) case will help me understand the general theory better.