In an infinite-dimensional Banach space $(X, \|\cdot\|)$ with a countable Schauder basis $\{x_n\}$, define:
$$ F_r: \operatorname{Span}(\{x_n\}) \rightarrow \operatorname{Span}(\{x_n\}), \hspace{0.3cm} \sum_{i\leq N} \lambda_i x_i \mapsto \sum_{i\leq N} \operatorname{Re}(\lambda_i) x_i $$
and:
$$ F_i: \operatorname{Span}(\{x_n\}) \rightarrow \operatorname{Span}(\{x_n\}), \hspace{0.3cm} \sum_{i\leq N} \lambda_i x_i \mapsto \sum_{i\leq N} \operatorname{Im}(\lambda_i) x_i $$
- Can $F_r$ be always continuously extended to the entire $X$?
Next, define another norm $\|\cdot\|_1$ in $\operatorname{Span}(\{x_n\})$ as $\|x\|_1 = \|F_r(x)\| + \|F_i(x)\|$ and let $X_1$ be the completion of $\operatorname{Span}(\{x_n\})$ with respect to $\|\cdot\|_1$. Notice that in the case when $F_r$ could be continuously extended, $F_i$ could be too and we could find $M>0$ such that $\|x\|_1 \leq M\|x\|$ for each $x\in X$. Then:
- In the case when $F_r$ (or $F_i$) could be continuously extended, can $\|\cdot\|$ an $\|\cdot\|_1$ be equivalent?
Similar to how the idea of proving the real version of the Hahn-Banach Theorem is extended to the complex version one, when we deal with problems in $X^{\ast}$, we could always start from a real bounded linear functional. However, I am not sure if, when dealing with problems related to Banach space geometry, it can be assumed what holds in a real Banach space will also hold in a complex one.