Editing Bra–ket notation (section)

===Notation===

Since kets are just vectors in a Hermitian vector space, they can be manipulated using the usual rules of linear algebra. For example:

:<math>\begin{align}
|A \rangle &= |B\rangle + |C\rangle \\
|C \rangle &= (-1+2i)|D \rangle \\
|D \rangle &= \int_{-\infty}^{\infty} e^{-x^2} |x\rangle \, \mathrm{d}x \,.
\end{align}</math>

Note how the last line above involves infinitely many different kets, one for each real number {{math|''x''}}.

Since the ket is an element of a vector space, a '''bra''' <math>\langle A|</math> is an element of its [[dual space]], i.e. a bra is a linear functional which is a linear map from the vector space to the complex numbers. Thus, it is useful to think of kets and bras as being elements of different vector spaces (see below however) with both being different useful concepts.

A bra <math>\langle\phi|</math> and a ket <math> |\psi\rangle</math> (i.e. a functional and a vector), can be combined to an operator <math>|\psi\rangle\langle\phi|</math> of rank one with [[outer product]]

:<math>|\psi\rangle\langle\phi| \colon |\xi\rangle \mapsto |\psi\rangle\langle\phi|\xi\rangle ~.</math>