Let $\Gamma$ be an uncountable set (possibly of cardinality $\aleph_1$). Is there an injective bounded linear operator $T\colon c_0(\Gamma)\to X$, where
a) $X$ is some separable Banach space
Thank you in advance.
EDIT: This might be useful as well: Johnson and Zippin proved that each quotient of $c_0$ is in fact its subspace. Is there a similiar result for general $c_0(\Gamma)$ spaces?
EDIT 2: Another hint: If $T_1\colon Y\to c_0(\Gamma)$ is injective, then there exists an injective operator $S\colon Y\to c_0(\Gamma)$ with dense range.