Editing Bra–ket notation (section)

===Associativity===
Given any expression involving complex numbers, bras, kets, inner products, outer products, and/or linear operators (but not addition), written in bra–ket notation, the parenthetical groupings do not matter (i.e., the [[associative property]] holds). For example:
:<math>\begin{align}
\lang \psi| \bigl(A |\phi\rang\bigr) = \bigl(\lang \psi|A\bigr)|\phi\rang \, &\stackrel{\text{def}}{=} \, \lang \psi | A | \phi \rang \\
\bigl(A|\psi\rang\bigr)\lang \phi| = A\bigl(|\psi\rang \lang \phi|\bigr) \, &\stackrel{\text{def}}{=} \, A | \psi \rang \lang \phi |
\end{align}</math>
and so forth. The expressions on the right (with no parentheses whatsoever) are allowed to be written unambiguously ''because'' of the equalities on the left. Note that the associative property does ''not'' hold for expressions that include nonlinear operators, such as the [[antilinear]] [[T-symmetry|time reversal operator]] in physics.