Are these two "Mazur's" theorems the same? According to Royden:

Mazur's Theorem: Let $K$ be a nonempty convex subset of a normed linear space $X$; $K$ is strongly closed if and only if it is weakly closed.

According to Wikipedia:

Mazur's Theorem: Let $M$ be a proper vector subspace of the topological vector space $X$ and suppose $K$ is a nonempty convex open subset of $X$ disjoint from $M$. Then there is a closed hyperplane $H\subset X$ containing $M$, but remaining disjoint from $K$.

I am not especially interested in the history or etymology of these theorems; I am really interested in whether or not they are fundamentally the same, reflections of each other. I say this because I have taken notes on the proof of Royden's version, but am currently at a loss when it comes to Wikipedia's version - if the two theorems are basically the same, I can have a go at proving Wikipedia's version myself, but if they turn out quite different it is likely that at this stage in my education I won't be able to prove Mazur's theorem (Wiki version) and I really don't want to waste my time...
Some thoughts:
For $M\subseteq H$ and $H\cap K=\varnothing$ we require a continuous $\psi\in X^\sharp$ with $H=\psi^{-1}(c)$ for some $c\in\Bbb R$: we need $\psi(m)=c$ for all $m\in M$ and $\psi(k)\neq c$ for all $k\in K$. As $M$ is convex and $K$ is open convex, one variant of the hyperplane separation theorem gives that there exists a continuous $\varphi\in X^\sharp$ such that $\varphi(k)\lt s\le\varphi(m)$ for all $k\in K,\,m\in M$ and some constant $s\in\Bbb R$, but these statements are not the same (although Wikipedia hints that there is some immediate connection). For $\varphi(m)\ge s$ for all $m$ yet we wish for $\varphi(m)=s$ for all $m$, which would require that $\varphi(m_1+m_2)=\varphi(m_1)+\varphi(m_2)=\varphi(m_1)\implies\varphi(m_2)=0$ for all $m_1,m_2\in M$, so we would like $M\subseteq\ker\varphi$ and $s=0$.
The kernel is immediately a closed hyperplane by continuity but Royden's edition of the theorem with $K$ closed is not apparently relevant.
 A: I would say that both theorems are different faces of the same fact. At their heart, both are separation theorems.

*

*The first theorem is typically proved the following way: If $K$ is (strongly) closed and convex and $x \in X \setminus K$, then you can find an affine closed hyperplane containing $x$ and disjoint from $K$. This implies that $K$ is weakly closed. (This also works in topological vector spaces).


*In the second theorem, it should be enough to consider the case $M = \{0\}$ (by going to a quotient space). Thus you want to separate the point $0$ from the open convex set $K \not\ni 0$ by a closed hyperplane.
Now, we can relate both theorems:

*

*If we believe in the second theorem, and $x \not\in K$ with $K$ closed and convex, then we can find an open, convex superset of $K$ not containing $x$. Thus, we can invoke the second theorem and $x$ can be separated from $K$.


*Let us replace the first theorem by: "We can separate closed convex sets $C$ from points $x \not\in C$" (which is the typical way to prove theorem 1; however, I do not see immediately whether this is implied by theorem 1). This implies the second theorem: If $K$ is closed and open, $0 \not\in K$, we could consider the closed convex sets
$$
C_n := \{x \in X \mid \forall \lVert{h}\rVert < 1/n : x + h \in K\}
.$$
These can be separated from $0$ by some continuous linear $\psi_n$, $\lVert\psi_n\rVert = 1$. Then one can check that weak-$\star$ accumulation points of $\psi_n$ do not vanish and separate $K$ from $0$.
A: They are just two different theorems (one about separating hyperplanes, the other on closed convex sets in the weak and strong topology) that have been proved by Mazur (who has shown a lot more theorems, see Wikipedia for his career).
There are also several theorems due to Cantor (so named),Brouwer, Banach, Luzin, Borel and many other early 20th century great mathematicians. There need not be a connection beyond that the common discoverer. Mazur was a pioneer in convex analysis, it's not surprising to have several theorems on convex sets with his name, IMO.
