Background and notations
Let $\mathcal{X}$ denote an input domain. Let $H$ denote the Reproducing Kernel Hilbert Space (RKHS) induced by the Radial basis function (RBF) kernel with bandwith 1, which is defined as the positive-definite kernel $$k(x,y) = \exp\left(-\frac{1}{2}\|x - y\|^2\right)$$
$H$ is space of functions from $\mathcal{X}\to \mathbb{R}$. The functions in $H$ satisfy the reproducing property, stated as follows $$(\forall f\in H) (\forall x\in \mathcal{X}) f(x) = \langle f, \phi(x)\rangle_{H}$$ where the feature map $\phi(x) : \mathcal{X}\to \mathbb{R}$ is defined as $$(\forall x'\in \mathcal{X})\;\phi(x)(x') := k(x,x')$$ Namely, the positive definite kernel $k$ induces a feature map $\phi$.
Using the reproducing property, we can now show that $H$ continuously embeds into $L^2(\pi)$, where $\pi$ is any probability measure on $\mathcal{X}$. We denote this embedding by $\iota$, and it's used in the statement of the question.
Indeed, we have $$\|\iota(f)\|_{L^2}^2 := \int_{\mathcal{X}}f(x)^2d\pi(x) \leq \int_{\mathcal{X}}\langle f, \phi(x)\rangle_{H}^2\;d\pi(x) \leq \|f\|_{H}^2$$ due to Cauchy-Schwarz inequality and the reproducing property for the RKHS, that allows us to express $$f(x) = \langle f, \phi(x)\rangle_{H},\quad k(x,x') = \langle \phi(x),\phi(x')\rangle_{H}$$ for any element $f$ of the RKHS, and the fact that $\langle \phi(x),\phi(x)\rangle_{H} = k(x,x)\leq 1$ due to the boundedness of the positive-definite kernel $k$.
Question
Consider the closed ball defined as follows $$B = \{\|f\|_{H}\leq 1\}$$
By definition $B$ is a closed ball with respect to the RKHS norm $\|\cdot \|_{H}$. Let $\iota$ denote the continuous embedding from $H$ to $L^2(\pi)$, which is shown to exist in the above section. Is the image $\iota(B)$ of $B$ under this embedding a closed set (w.r.t. the $L^2(\pi)$-topology)?
