# This month in mathlib (Jun 2022)

We apologize for the delay in posting this overview. In June 2022 there were 460 PRs merged into mathlib. We list some of the highlights below.

• Functional analysis

• PR #14135 proves compactness of the character space of a normed algebra. This is a key step in developing the continuous functional calculus.
• PR #8112 proves the Krein-Milman theorem for real normed spaces: any compact convex set is the closure of the convex hull of its extreme points.
• PR #14677 proves Stone's separation theorem: in a vector space over a linearly ordered field, for any pair of disjoint convex subsets, one can partition the whole space into two convex subsets containing the given sets. This is a generalization of the geometric Hahn-Banach theorem where the partition is made of half-spaces.
• Measure theory and probability

• PR #14578 characterizes weak convergence of finite measures in terms of integrals of bounded continuous real-valued functions.
• PR #14755 introduces moments and moment generating function of a real random variable.
• PR #13592 solves the Ballot problem: if in an election, candidate A receives $p$ votes whereas candidate B receives $q$ votes where $p > q$ then the probability that candidate A is strictly ahead throughout the count is $(p - q) / (p + q)$.
• Model theory

• Number theory and group theory

• PR #14332 defines the ramification index and inertia degree of ring extensions.
• PR #14312 defines the p-primary component of a group.
• PR #13524 gives the classification of finitely generated torsion modules over a PID and PR #14736 specializes this to finite abelian groups.
• Linear algebra

• PR #14231 reformulates the spectral theorem in terms of matrices.
• Tactics

• The linear_combination tactic became more powerful in PR #14229 by allowing combinations of arbitrary proofs.