(25) |

for all ket vectors and , and

(26) |

for all complex numbers . Operators and are said to be equal if

(27) |

for all kets in the ket space in question. Operator is termed the

(28) |

for all ket vectors in the space. Operators can be added together. Such addition is defined to obey a commutative and associate algebra: i.e.,

(29) | ||

(30) |

Operators can also be multiplied. The multiplication is associative:

(31) | ||

(32) |

However, in general, it is

(33) |

So far, we have only considered linear operators acting on ket vectors. We can also give a meaning to their operation on bra vectors. Consider the inner product of a general bra with the ket . This product is a number that depends linearly on . Thus, it may be considered to be the inner product of with some bra. This bra depends linearly on , so we may look on it as the result of some linear operator applied to . This operator is uniquely determined by the original operator , so we might as well call it the same operator acting on . A suitable notation to use for the resulting bra when operates on is . The equation which defines this vector is

for any and . The triple product of , , and can be written without ambiguity, provided we adopt the convention that the bra vector always goes on the left, the operator in the middle, and the ket vector on the right.

Consider the dual bra to
. This bra depends antilinearly on
and must therefore depend linearly on
.
Thus, it may
be regarded as the result of some linear operator applied to
.
This operator is termed the *adjoint* of
, and is denoted
. Thus,

It is readily demonstrated that

(36) |

plus

(37) |

It is also easily seen that the adjoint of the adjoint of a linear operator is equivalent to the original operator. An

(38) |