# What is the natural norm on these spaces?

In [1] the authors define the function spaces \begin{align*} V(\mathbb{R}^N) = \lbrace v \in \mathbb{R}^N \to \mathbb{C}: ~ &\nabla v \in L^2(\mathbb{R}^N), \\ &\Re(v) \in L^2(\mathbb{R}^N), \\ & \Im(v) \in L^4(\mathbb{R}^N), \\ &\nabla \Re(v) \in L^{4/3}(\mathbb{R}^N) \rbrace \end{align*} and $$W(\mathbb{R}^N) = \lbrace 1 \rbrace + V(\mathbb{R}^N).$$

Later in the same paper they claim that the functional $$v \mapsto \int_{\mathbb{R}^N} (1-\vert 1+v \vert^2)^2$$ is continuous in $V(\mathbb{R}^N)$ and thefore the functional $$w \mapsto \frac{1}{2} \int_{\mathbb{R}^N} \vert \nabla w\vert^2 +\frac{1}{4}\int_{\mathbb{R}^N} (1-\vert w \vert^2)^2$$ is continuous in $W(\mathbb{R}^N)$.

I would like to check this, but I don't even know what the norm on these spaces is supposed to be. So my question is:

What norm do the authors implicitly equip these spaces with to infer the continuity of the above functionals and how do you start checking the continuity?

One idea would be $$\Vert v \Vert_V = \Vert \Re(v) \Vert_{L^2} + \Vert \nabla v \Vert_{L^2} + \Vert \Im(v) \Vert_{L^4} + \Vert \nabla \Re(v) \Vert_{L^{4/3}}$$ which indeed is a norm. But what about the norm on $W$?

[1] Béthuel, F., P. Gravejat und J. C. Saut: Travelling waves for the Gross- Pitaevskii equation. II. Comm. Math. Phys., 285(2):567–651, 2009.

Finally, the space $W(\mathbb{R}^n)$ is not a vector space, but rather an affine space. Hence you may just notice work with those elements of the form $1+v$ with $v \in V(\mathbb{R}^N)$, and you get an induced topological structure by declaring that $1+v_1$ is "close" to $1+v_2$ if and only if $v_1$ is "close" to $v_2$.
• Thank you very much! But how does one prove the continuity? For simplicity consider the functional $F(v) = \int \vert Dv \vert^2$. Now if $\Vert v-w \Vert_V < \delta$ in particular we have that $\Vert D v - Dw \Vert_{L^2} < \delta$. But $\vert F(v)-F(w) \vert = \vert \int \vert Dv \vert^2 - \vert Dw \vert^2 \vert$. How does one proceed? Any hint? – mjb Jul 25 '13 at 7:36