Hyperplane separation theorem

Misunderstanding is crucial, when their connection is empty. This abstract algebra is called Krull's separation lemma.

I∩M=∅

I is ideal, and M is multiplicative and closed.

P is the prime ideals for the integers that contain all the multiples of a given prime number, together with the zero ideal.

I⊆P

P∩M=∅

This is disjoint convex sets in higher dimensional Euclidean space. A and B are disjoint nonempty convex subsets.

[x,v]≧c and [y,v]≦c

v is a nonzero vector, and c is a real number. x is in A, and y is in B. If both sets are closed, and at least one of them is compact, then the separation can be strict. This is called Hyperplane separation theorem.