Sesquilinear Form - Definition and Conventions

Definition and Conventions

Conventions differ as to which argument should be linear. We take the first to be conjugate-linear and the second to be linear. This is the convention used by essentially all physicists and originates in Dirac's bra-ket notation in quantum mechanics. The opposite convention is perhaps more common in mathematics but is not universal.

Specifically a map φ : V × V → C is sesquilinear if

for all x,y,z,w ∈ V and all a, b ∈ C.

A sesquilinear form can also be viewed as a complex bilinear map

where is the complex conjugate vector space to V. By the universal property of tensor products these are in one-to-one correspondence with (complex) linear maps

For a fixed z in V the map is a linear functional on V (i.e. an element of the dual space V*). Likewise, the map is a conjugate-linear functional on V.

Given any sesquilinear form φ on V we can define a second sesquilinear form ψ via the conjugate transpose:

In general, ψ and φ will be different. If they are the same then φ is said to be Hermitian. If they are negatives of one another, then φ is said to be skew-Hermitian. Every sesquilinear form can be written as a sum of a Hermitian form and a skew-Hermitian form.