Case:
which reminds me of some version of the Hahn-Banach theorem. So I think we need to prove the sublinearity i.e. positive homogeneity and the subadditivity.
My Proposal #2 based on Svetoslav's suggestion
Corollary 3.1.7/p.48 from the book "Functional analysis (An introduction)" by Yuli Eidelman, Vitali Milman, Antonis Tsolomitis - converted to this case:
Let $X$ be a normed $\mathbb{C}$-vector space. Let $v \in X \setminus Z$ and $Z \subset X$ is a closed subspace of a normed space $X$, meaning the distance \begin{equation*} d = dist(v, Z) > 0. \end{equation*} Then, there exists $\Phi \in X'$ such that $\Phi(v) \neq 0$ and $\Phi(u)=0$ for all $u \in Z$ ( meaning $\Phi(Z)=0$ or $Z\subset Ker (\Phi)$ ).
Proof: First consider $Z_{\Phi} = span(v, Z)$: \begin{equation*} Z_{\Phi} = \{ \lambda v + u: \lambda \in \mathbb{R}, u \in Z \}. \end{equation*} Let \begin{equation*} \Phi(z) := \lambda. \end{equation*} where the function $\Phi(z)$ is well defined because $z = \lambda v + u$ can be written in a unique way: $\lambda \in \mathbb{R}$ and $u \in Z$ are defined uniquely by $z$: Indeed, if $z=\lambda_1v+u_1=\lambda_2v+u_2$, where $u_1, u_2\in Z$, then $(\lambda_1-\lambda_2)v + 1\cdot (u_1-u_2)=0$. Now, if $\lambda_1=\lambda_2\Rightarrow u_1=u_2$. If $\lambda_1\neq \lambda_2\Rightarrow v=-\frac{u_1-u_2}{\lambda_1-\lambda_2}\in Z$ which is a contradiction with that $v\notin Z$
The linearity of this function: let $\mu\in\mathbb K$ $z_1,z_2,z\in Z_{\Phi}$ where $z_1=\lambda_1 v+ u_1$, $z_2=\lambda_2 v+u_2$, $z=\lambda v+u$, $u_1,u_2,u\in Z$. 1) $$\Phi(z_1+z_2) =\Phi(\lambda_1 v+u_1 +\lambda_2 v+ u_2)=\Phi((\lambda_1+\lambda_2)v +(u_1+u_2))$$ which by the definition of $\Phi$ equals $\lambda_1+\lambda_2=\Phi(z_1)+\Phi(z_2)$.
2) $$\Phi(\mu z)=\Phi(\mu(\lambda v+u))=\Phi(\mu\lambda v+\mu u)=\mu\lambda$$ because $\mu u\in Z$
Now, it is clear that $\Phi(u)=\Phi(0\cdot v+u)=0,\,\,\forall u\in Z$, so $\Phi(Z)=0$. Also $\Phi(v)=\Phi(1\cdot v + 0)=1\neq 0$. By Hahn Banach theorem we can extend the linear bounded functional $\Phi$ defined on the linear subspace $Z_{\Phi}$ to a linear and bounded functional $\bar \Phi$ with the same norm on the whole space $X$.
How can you prove rigorously the Complex Hahn-Banach theorem?