Question

For $a \in \mathbb{R}^N\setminus{0}, N \ge 2$, and $\lambda \in \mathbb{R}$ let $$ X_{\lambda,a}=\{u(\cdot+\lambda a):\, u(x)=u(|x|) \in W^{1,2}(\mathbb{R}^N)\}. $$ Denote by $X_a$ the closure of the direct sum: $$ \bigoplus_{\lambda \in \mathbb{R}}X_{\lambda,a}. $$ Question: Is $X_a$ a proper subspace of $W^{1,2}(\mathbb{R}^N)$?

Answer 1

Yes, all functions in $X_a$ are still symmetric wrto the line generated by $a$. And all traces of these functions on the affine hyperplanes orthogonal to $a$ are radially symmetric in dimension $N-1$.

edit. Consider any pair $x$ and $x'$ in $\mathbb{R}^N$ with $|x|=|x'|=1$ and $a\cdot x=a\cdot x'$. Then $|x-\lambda a |=|x'-\lambda a|\ ,$ so $u(x)=u(x')$ for all $\lambda$ and all $u\in X_{\lambda,a}$. And this symmetry is preserved taking the linear span and the closure: $u=u\circ R$ holds for any $u\in X_a$ for any orthogonal $R$ that fixes $a\ .$