I'm reading a proof in trying to prove that a Banach space $X$ is reflexive if and only if $X^{*}$ is reflexive. There's a point in the proof saying that $X$ is a closed subspace of $X^{**}$, but I don't know why. We can define an embedding from $X$ to $X^{**}$ by the mapping
$$C: X \rightarrow X^{**} $$ $$x \rightarrow g_{x}$$ here $g_{x}(f) = f(x)$
This mapping is isometry an an isomorphism from $X$ onto $R(C)$. Also, I see many times somw authors say that if we define a linear continuous mapping $C:X \rightarrow Y$, then $R(T)$ is closed subspace in $Y$, so I wonder if this is always true for every normed space $X$ and $Y$. If not, can we add some conditions to make it always true? Can anyone help me clarify this? Thanks. I really appreciate.
Image of every bounded below (in particular isometric) operator from Banach space to a normed space have closed range. But in general this is not true. For example natural embedding of $c_{00}$ into $c_0$ is isometric but its range is not closed because $c_{00}$ is dense in $c_0$.