In mathematics, a sesquilinear form is a generalization of inner products of complex vector spaces, which are the most common sesquilinear forms. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the name; which originates from the Latin numerical prefix sesqui- meaning "one and a half". The basic concept of inner products – producing a scalar from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of a vector.

A motivating special case is a sesquilinear form on a complex vector space, V. This is a map V × V → C that is linear in one argument and "twists" the linearity of the other argument by complex conjugation (referred to as being antilinear in the other argument). This case arises naturally in mathematical physics applications. Another important case allows the scalars to come from any field and the twist is provided by a field automorphism.

An application in projective geometry requires that the scalars come from a division ring (skew field), K, and this means that the "vectors" should be replaced by elements of a K-module. In a very general setting, sesquilinear forms can be defined over R-modules for arbitrary rings R.

Sesquilinear forms abstract and generalize the basic notion of a Hermitian form on complex vector space. Hermitian forms are commonly seen in physics, as the inner product on a complex Hilbert space. In such cases, the standard Hermitian form on Cⁿ is given by

where ${\displaystyle {\overline {w}}_{i}}$ denotes the complex conjugate of ${\displaystyle w_{i}~.}$ This product may be generalized to situations where one is not working with an orthonormal basis for Cⁿ, or even any basis at all. By inserting an extra factor of ${\displaystyle i}$ into the product, one obtains the skew-Hermitian form, defined more precisely, below. There is no particular reason to restrict the definition to the complex numbers; it can be defined for arbitrary rings carrying an antiautomorphism, informally understood to be a generalized concept of "complex conjugation" for the ring.

Conventions differ as to which argument should be linear. In the commutative case, we shall take the first to be linear, as is common in the mathematical literature, except in the section devoted to sesquilinear forms on complex vector spaces. There we use the other convention and take the first argument to be conjugate-linear (i.e. antilinear) and the second to be linear. This is the convention used mostly by physicists and originates in Dirac's bra–ket notation in quantum mechanics. It is also consistent with the definition of the usual (Euclidean) product of ${\displaystyle w,z\in \mathbb {C} ^{n}}$ as ${\displaystyle w^{*}z}$.

In the more general noncommutative setting, with right modules we take the second argument to be linear and with left modules we take the first argument to be linear.

Over a complex vector space ${\displaystyle V}$ a map ${\displaystyle \varphi :V\times V\to \mathbb {C} }$ is sesquilinear if