Has there ever been a substantial change in notation resulting from a landmark theorem?

Question

Often times the notation we use for mathematical concepts is based in part on properties of that concept. For example, the notation $|A| = |B|$ in set theory means there’s a bijection between $A$ and $B$, and the notation is warranted because the “there is a bijection between these two sets” relation is an equivalence relation. The notation $A \le_p B$ for polynomial-time reducibility works both because of the intuitive connection between the hardness of $A$ and $B$ and because reducibility is reflexive and transitive.

Have there been any major examples of a concept being well-known and studied with one set of notation whereupon a surprising or fundamental theorem was proved that caused folks to revisit the old notation and swap it out for newer notation based on the insights of the theorem?

Answer 1

Bra-ket notation?

I believe some use bra-ket notation as implying certain things like being strictly for wavefunctions or implying normalization, but in reality it is just a vector notation which has two main advantages:

• it defines an explicit notation for dual vectors, making things like the Dirac's delta being well defined in this notation (it is a vector in the dual vector space that is not dual to any vector in the original space)
• it makes it easy to define vectors via the eigenvalues of commuting operators. This is just because it is the only vector notation that encloses the name of the vector rather than changing font or adding something at the top or the bottom. It's similar to how XML tags are convenient because they allow to add properties inside the tag.

I would say that the discovery of quantum mechanics just made the use of dual vectors along side normal vectors so standard that eventually it forced to fix the lacking of the original vector notations. You can fix the first point also in the usual vector notation with the arrow on top by assigning to dual vectors the arrow in the opposite direction. The second point however does require and enclosing notation, so eventually a new notation would have been needed anyway.