Complex Conjugate Vector Space - Antilinear Maps

Antilinear Maps

If and are complex vector spaces, a function is antilinear if

for all and .

One reason to consider the vector space is that it makes antilinear maps into linear maps. Specifically, if is an antilinear map, then the corresponding map defined by

is linear. Conversely, any linear map defined on gives rise to an antilinear map on .

One way of thinking about this correspondence is that the map defined by

is an antilinear bijection. Thus if if linear, then composition is antilinear, and vice versa.