We shall use n.v.s to refer to normed vector space.
Definition 1 Let be a n.v.s. A set of the form is called a hyperplane.
Definition 2 We call a set of the form a half-space determined by . Replacing with gives the other half-space.
Proposition 1 Any half-space is a convex set.
Proposition 2 Let be a n.v.s. such that . For any , the hyperplane is closed iff is continuous.
Proof: open iff is closed. Suppose is continuous and that for , . Take some open neighborhood of not containing . Then, is open and disjoint with .
To prove that is continuous, it suffices to prove that is bounded, by Proposition 2 of . To show that is bounded, showing that suffices, where is the unit ball centered at zero. To do so, it suffices to show that every such that has an open neighborhood of the form such that for any , , since if this holds, then for ,
Assume that is open. Then, every such that is contained by a . Suppose that satisfies . Then, there must exist some such that , a contradiction. This completes our proof.
Definition 3 Let be two subsets of and with . If is such that of its two half-planes, one contains and the other contains , then we say that separates and . We say that strictly separates and if there exists some such that
We now wish to prove that for any open convex set containing and any , there exists a hyperplane that separates and . Let denote the linear functional associated with this separation. We prescribe for some . If we can find some sublinear function that is strictly bounded above on by such that on the subspace satisfies , then we can apply the Hahn-Banach theorem to extend to all in order separate from by a hyperplane.
For this desired sublinear function , we try the following.
Definition 4 Let be a subset of n.v.s . Then the gauge or Minkowski functional with respect to is defined by .
Proposition 3 If contains an open ball centered at , the Minkowski functional satisfies the condition that for all , .
Proof: Trivial and left to the reader.
We want that on , , which is satisfied when we let . We let . That is open means there is some open ball centered at the origin . For any , obviously holds. This is a simple stupid way to uniformly upper bound on .
Proposition 4 For any subset , is bounded above by for . If is open, then for all , .
Proof: Trivially proven.
Proposition 5 If is an open convex set containing , then .
Proof: Since we can scale down arbitrarily by Proposition 3, it suffices to prove this triangle inequality on an open ball centered at that is contained by . Moreover it suffices to prove that for any ,
is satisfied. We notice that
Let . Obviously . With this in mind, that is convex implies that for any for any , . We use to set the value of noticing that we can multiply by the denominator to derive the desired inequality. We calculate
We notice that if we replace with , we would obtain the desired inequality. Because is convex must be connected. Assuming that , we have . The connectedness hypothesis implies in fact that all such that must be in . We have that
which then implies for . Thus, we are able to make the aforementioned replacement, which then completes our proof.
Proposition 6 If is an open convex set containing , then its associated Minkowski functional is a subadditive function.
Proof: The two requirement of subadditive were proven in and respectively.
Theorem 1 Let be an open convex subset of a n.v.s . For arbitrary , there exists such that for all . In particular, the hyperplane separates and .
Proof: After a translation, we may always assume that . The Minkowski functional by Proposition 6 is a subadditive function. We define on to be linear. We can extend to on all of by the Hahn-Banach theorem (see  for details on it). Since by Proposition 4, for any , , we have for all , . Thus, setting gives us what we want to prove.
In , propositions of more general separation of sets by hyperplanes were proven. I shall not write them up here because I believe the foundational ideas behind their proofs have already been explained in the propositions above. I had learned of the Minkowski functional this week but initially did not feel like I grasped the motivation behind it. Such difficulty was resolved after reading 1.2 in , the title of which is “The Geometric Forms of the Hahn-Banach Theorem: Separation of Convex Sets”. Interestingly, I read about the Minkowski functional on Wikipedia before writing up , in the process of which I gained non-trivial understanding of the Hahn-Banach Theorem, which I had learned in 2017 or 2018 but forgotten in 2021 due to more or less superficial understanding. Certainly, the Hahn-Banach theorem can come across as very formal and abstract at first encounter, and it might not be immediate why it’s so significant. However, the geometric interpretation of it via separation of convex sets gives it more concrete meaning.
-  gmachine1729：On normed vector spaces
-  gmachine1729：How to interpret the Hahn-Banach theorem
-  Haim Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Science+Business Media, LLC, 2011