physical description

colorSpec
type

infinite dimensional
vector space description

vector
space symbol

finite dimensional description

a light source,
this includes both physical and ideal sources

light

the space of absolutely integrable functions on an interval of wavelengths, e.g. [380,780] nm. For simplicity, we denote this space L1[380,780] by L (for light).

L

row vectors

a light responder,
also called a detector

responsivity.light

the space of continuous functionals on L, which is the dual space L*. This dual of L1[380,780] is isomorphic to L[380,780] = the essentially bounded functions on [380,780]. All such functionals are given by an inner product with a function in L[380,780]

L*

column vectors

a diffuse reflector,
or a non-diffuse transparent material

material

the space of multiplication operators on L , which is denoted by M (for material or multiplier) and is isomorphic to L[380,780].

M

diagonal matrices

a material responder

it is typically defined by a fixed light source and fixed light responder, with a variable material between them,
i.e. a
scanner

responsivity.material

the space of continuous functionals on M , which is the dual space M*. It is not isomorphic to L1[380,780]; M* is bigger.

M*

diagonal matrices

Table 1.1 The 4 types of spectra and their corresponding vector spaces