If $X$ is inseparable, must $Y$ be inseparable?

Question

Let $X$ and $Y$ be normed vector spaces. Suppose that there is linear isometry $T: X \rightarrow Y$, namely that $\Vert Tx\Vert =\Vert x\Vert$ for all $x\in X$.

If $X$ is inseparable, must $Y$ be inseparable?

And if $Y$ is inseparable, must $X$ be inseparable?

Answer 1

$X=\textrm{c}_{00}$ and $Y=\ell_{\infty}$ both equipped with $\|⋅\|_{\infty}$ norm.

Consider $T:X\to Y$ defined by $Tx=x$

Then $T$ is a linear isometry.

$(\textrm{c}_{00}, \|⋅\|_{\infty})$ is separable but $(\ell_{\infty}, \|⋅\|_{\infty}) $ is not separable.

$Y$ non separable doesn't imply $X$ non separable.

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But the other way is true. $X$ is not separable implies $Y$ is not separable. Contrapositively $Y$ separable implies $X$ separable.

$T:X\to Y$ linear isometry implies $X$ is isometrically isomorphic to $T(X) \subset Y$. Now if $Y$ is separable then $T(X) $ is separable implies $X$ separable.

$X$ non separable implies $Y$ non separable.