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For $a \in \mathbb{R}^N\setminus{0}$, 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 $W^{1,2}(\mathbb{R}^N)$?

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Subspaces of a Sobolev space

For $a \in \mathbb{R}^N\setminus{0}$, 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 $W^{1,2}(\mathbb{R}^N)$?