**Other References Containing the Convex Hahn-Banach**

The convex majorant version of HBT (Hahn-Banach Theorem) also appears in

**"Convex" Hahn-Banach for Complex Vector Spaces**

Reed and Simon also have a "convex" majorant HBT for complex vector spaces. The standard HBT for complex spaces has a semi-norm $p$ and a complex-linear functional $f$ with $|f| \leq p$. In the ``convex" majorant HBT for complex spaces, $p$ need not be a semi-norm, but is only required to satisfy the weaker condition
\begin{align}\label{1}\tag{1}
p(\alpha x + \beta y) \leq |\alpha|p(x) + |\beta|p(y) \quad \text{ when } |\alpha|+|\beta|=1.
\end{align}

Reed and Simon deduce the "convex" complex HBT from the convex real HBT in the same way that the standard semi-norm complex HBT is deduced from the standard sublinear real HBT (that is, by exploiting the relationship between a complex-linear functional and its real part). And, of course, the proof of the convex real HBT is basically identical to the proof of the sublinear real HBT.

Alternatively, the "convex" complex HBT can be deduced from the semi-norm complex HBT by changing Fedja's $P(x)$ to $P(x)=\inf_{0 \neq s \in \mathbb{C}} |s|^{-1}p(sx)$. Indeed, if $p$ satisfies \eqref{1}, then $P$ is a semi-norm; if a linear functional is dominated by $p$, it is also dominated by $P$; and $P \leq p$.

**Who First Noted the Convex HBT**

Since the proofs of the convex and sublinear HBT are so similar and in light of Fedja's argument, I would guess that the convex HBT was known basically as soon as the sublinear HBT was. That being said....

Schechter on page 318 points out that the convex HBT is not mentioned much in the literature but was known at least as early as in

See also

According to the last reference, the convex real HBT may also appear in the following two references, but I haven't been able to check.

H. Nakano: Modulared linear spaces, Jour. Fac. Sci. Univ. Tokyo, I, 6, 85 (1951).

H. Nakano: Topology and Linear Topological Spaces, Tokyo (1951).

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