(resp. by the set of all continuous linear (resp. continuous antilinear) functionals on which is called the (resp. the ) of If then linear functionals on are the same as antilinear functionals and consequently, the same is true for such continuous maps: that is, One-to-one correspondence between linear and antilinear functionals Given any functional the is the functional This assignment is most useful when because if then and the assignment reduces down to the identity map. The assignment defines an antilinear bijective correspondence from the set of all functionals (resp. all linear functionals, all continuous linear functionals ) on onto the set of all functionals (resp. all linear functionals, all continuous linear functionals ) on Mathematics vs. physics notations and definitions of inner product The Hilbert space has an associated inner product valued in 's underlying scalar field that is linear in one coordinate and antilinear in the other (as described in detail below). If is a complex Hilbert space (meaning, if ), which is very often the case, then which coordinate is antilinear and which is linear becomes a important technicality. However, if then the inner product is a symmetric map that is simultaneously linear in each coordinate (that is, bilinear) and antilinear in each coordinate. Consequently, the question of which coordinate is linear and which is antilinear is irrelevant for real Hilbert spaces. Notation for the inner product In mathematics, the inner product on a Hilbert space is often denoted by or while in physics, the bra–ket notation or is typically used instead. In this article, these two notations will be related by the equality: Competing definitions of the inner product The maps and are assumed to have the following two properties: The map is in its coordinate; equivalently, the map is linear in its coordinate. Explicitly, this means that for every fixed the map that is denoted by and defined by is a linear functional on In fact, this linear functional is continuous, so The map is linear in its coordinate; equivalently, the map is linear in its coordinate. Explicitly, this means that for every fixed the map that is denoted by and defined by is an antilinear functional on In fact, this antilinear functional is continuous, so In mathematics, the prevailing convention (i.e. the definition of an inner product) is that the inner product is coordinate and antilinear in the other coordinate. In physics, the convention/definition