If $(X,\|.\|_*)$ is complete and $\|.\|_{*} \leq \|.\|_{**}$, then is $(X,\|.\|_{**})$ complete?

Question

If a space $X$ is complete with respect to $\|.\|_*$ and we have that $\|f\|_{*} \leq \|f\|_{**}$ for all $f \in X$.

Does this imply that the space will be also complete with respect $\|.\|_{**}$

Answer 1

Using a Hamel basis argument we can show that any infinite dimensional Banach space contains a discontinuous linear functional $f$ such that for some sequence $x_n \to 0$ we have $f(x_n) \to 1$. Let $\|.\|_{*}$ be the original norm and $\|.\|_{**}=\|.\|_{*}+|f(x)|$. Then the hypothesis is satisfied. But $\{x_n\}$ is a Cauchy sequence in the second norm which does not converge. [Let $\{y_n\}$ be linearly independent with $\|y_n\|=1$ for all $n$. Define $f(y_n)=n$ and extend $f$ linearly to $X$. Then $f$ is not continuous. Also, $\frac {y_n} n \to 0$ and $f(\frac {y_n} n) =1 \to 1$. Take $x_n=\frac {y_n} n $ in above argument].