Let $C_0$ be the Banach space of all continuous real value functions whose limits in $\pm \infty$ is zero, with the supremum norm. Is this space separable?
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There are continuous bijections between $\Bbb R$ and $(-1, 1)$. One example is $$ h(x) = \frac{2}{\pi}\arctan(x) $$ This means there is an isometry between your space and the space of continuous, real valued functions $f$ on $(-1, 1)$ with $\lim_{x \to -1^+}f(x) = \lim_{x \to 1^-} f(x)= 0$ (still with sup norm). The set of polynomials with rational coefficients (and with $(x^2 - 1)$ as factor) are dense on this space.
Arthur
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Alternatively, here is a countable dense set: Functions $$ p(x)e^{-x^2} $$ where $p$ is a polynomial with rational coefficients.
GEdgar
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