So I have this definition of Kadec norm

$\textbf{Definition:}$ Let $(X,\|\|)$ be a Banach space. The norm $\|\|$ is said to be a Kadec norm if $x_n \xrightarrow{w} \bar{x}$ and $\|x_n\|\to \|\bar{x}\|$ implies $x_n\to \bar{x}.$

I was wondering why is it necessary in this definition that $X$ to be a Banach space. I though that it is not at all, and that then the condition would imply that $X$ is Banach. But I was not able to prove that statement. Any thoughts?

No need for $X$ to be a Banach space. For example in Megginson's An introduction to Banach space theory, the Kadets-Klee property is defined for normed spaces (Definition 2.5.26, pg. 220). The same page also provides examples of non Banach spaces which have the Kadets-Klee property:
• The main question is: Why do we need to assume that $X$ is Banach? The property makes sense even without this assumption. I thought that this case was similar to reflexivity, where the condition already implies the completeness of the space – John D Jul 31 '17 at 1:51