The cones come in three types, with responses that peak at wavelengths corresponding to red, green,and blue light, respectively; see figure 1.
Hence two colors can look the same to a human observer but have a different spectral decomposition, which is the phenomenon of metamerism. This is a good thing in the sense that color output systems (such as computer monitors) exploit metamerism to reproduce color.
1) Once again you have overlooked the central issue and reversed the direction of implication.
I am tired of repeating this - could you please write it down and post it for me every time you post that muddle?
-> several different colors can be produced by one and the same spectral "color" combination (wavelength decomposition and recombination). The optical "illusion" link I posted demonstrates that.
-> The distances between colors do not correspond to the distances between points on the spectrum even, much less whatever way you imagine distances between non-spectral colors are measured. The wavelengths are not even in the same order as the colors in many common circumstances. The intransivity of the colors red, yellow, and violet demonstrate that.
So you can define a distance function over any color space; once you have at least two fixed wavelengths you automatically have a distance between them.
Again: You don't have two fixed colors, you have two fixed wavelengths - each of which can represent several different colors depending on circumstances, each of which can be produced by many different combinations of wavelengths (plus black and white).
Meanwhile: Even for the chosen wavelength combinations you need at least three, plus a way of making black and white.
Furthermore: You like Wiki? Ponder:
https://en.wikipedia.org/wiki/Visible_spectrum
Color displays (e.g.
computer monitors and
televisions) cannot reproduce
all colors discernible by a human eye. Colors outside the color
gamut of the device, such as most
spectral colors, can only be
approximated.
Approximations will of course not fulfill the requirements of a vector space or Hilbert space - so no vector space or Hilbert space involving the wavelength basis engineers use for color screen design and lens processing and so forth is possible for the colors of the perceived world.
How do you tell the difference between two colors with different single wavelengths? According to you, nobody can do this.
? Humans tell the difference between colors by looking at them. I never said humans were unable to perceive differences between colors.
All I said was that color space is not a Hilbert space (or even a vector space). I posted the reasons (lacks additivity, does not support an "inner product" or "distance" function, etc). And nothing you post about wavelengths is going to make any difference to that claim or its relevance here because (drum roll)
The thread is about colors.
Wavelengths are not colors.