One example of such objects is wavefunction of a particle *(x) in quantum physics. Strictly speaking set of possible wavefunctions do not form a Hilbert space because wavefunction has to normalized to 1. fiwcx))dx = 1. This means that we cannot multiply wavefunction by a scalar - the result will have a different norm, and, thus, will not describe a quantum particle. But we forget about this and treat y(x) as an element of a Hilbert space (the normalization condition imposes restriction on possible form of Schrodinger equation). In this project, your task will be to build a Hilbert space from usual colors. Colors form kind of a three-dimensional vector space in the sense that any color can be produced by mixing three "basis colors" (red). Igreen), and (blue) (standard computer RGB encoding). However, usual operations we perform with colors do not satisfy Hilbert axioms. For example, "blending two colors looks similar to addition, but blending color with itself does not change the color, (red) [blend with] Tred) = Tred), while Hilbert space addition should satisfy Tred) + (red) = 2 |red) # Tred). So, your main goal will be to define operation of addition of two colors, multiplication of a color by a real scalar, and operation of scalar product of two colors. These operations should satisfy all the axioms of the Hilbert spaces. Using these definitions, you will be able to do all the stuff we did with usual Hilbert spaces, for example, define linear operators and solve eigenvalue problems. Each color can be written as set of three numbers in standard RGB encoding. Icolor) - (R.G.B). For usual "physical" colors, each of the numbers R, G, and B varies between 0 and 1. RGB codes for some common colors are provided in the table below. Color Black White Red Green Blue Cyan RGB code (0, 0, 0) (1. 1. 1) (1. 0, 0) (0. 1, 0) (0. 0. 1) (0. 1. 1.) Magenta Yellow Gray Brown Orange Pink Purple (1, 0, 1) (1, 1, 0) (0.5, 0.5, 0.5) (0.6.0.4, 0.2) (1.0, 0.5, 0.0) (1.0, 0.5, 0.5) (0.5, 0.0, 0.5) 2 Define operations of addition of two colors. Icolor) + Icolorz), multiplication of a color by a real scalar, x |color). and scalar product of two colors (color, color) These operations must satisfy all axioms of Hilbert spaces and should be more-or-less intuitive, eg. addition should work similar to blending (yellow) + light blue) - Igreen)). "Define operation" means give description of the operation in terms of (R.G, B) codes of the color. For example, if color ) = (R, G., B.), Icolorz) = (R2.G2, B2), and Icolor,) + colorz) = |colorz) = (R.G.B). then you have to give formulas for Rs, G3, B, in terms of R.2.1.2. and B.2. R = Fr(R.G, B., R2, G2,B). G = F(R.G.B, R2.G.B.). B = F(R.G.B, R2, G2,B). One example of such objects is wavefunction of a particle *(x) in quantum physics. Strictly speaking set of possible wavefunctions do not form a Hilbert space because wavefunction has to normalized to 1. fiwcx))dx = 1. This means that we cannot multiply wavefunction by a scalar - the result will have a different norm, and, thus, will not describe a quantum particle. But we forget about this and treat y(x) as an element of a Hilbert space (the normalization condition imposes restriction on possible form of Schrodinger equation). In this project, your task will be to build a Hilbert space from usual colors. Colors form kind of a three-dimensional vector space in the sense that any color can be produced by mixing three "basis colors" (red). Igreen), and (blue) (standard computer RGB encoding). However, usual operations we perform with colors do not satisfy Hilbert axioms. For example, "blending two colors looks similar to addition, but blending color with itself does not change the color, (red) [blend with] Tred) = Tred), while Hilbert space addition should satisfy Tred) + (red) = 2 |red) # Tred). So, your main goal will be to define operation of addition of two colors, multiplication of a color by a real scalar, and operation of scalar product of two colors. These operations should satisfy all the axioms of the Hilbert spaces. Using these definitions, you will be able to do all the stuff we did with usual Hilbert spaces, for example, define linear operators and solve eigenvalue problems. Each color can be written as set of three numbers in standard RGB encoding. Icolor) - (R.G.B). For usual "physical" colors, each of the numbers R, G, and B varies between 0 and 1. RGB codes for some common colors are provided in the table below. Color Black White Red Green Blue Cyan RGB code (0, 0, 0) (1. 1. 1) (1. 0, 0) (0. 1, 0) (0. 0. 1) (0. 1. 1.) Magenta Yellow Gray Brown Orange Pink Purple (1, 0, 1) (1, 1, 0) (0.5, 0.5, 0.5) (0.6.0.4, 0.2) (1.0, 0.5, 0.0) (1.0, 0.5, 0.5) (0.5, 0.0, 0.5) 2 Define operations of addition of two colors. Icolor) + Icolorz), multiplication of a color by a real scalar, x |color). and scalar product of two colors (color, color) These operations must satisfy all axioms of Hilbert spaces and should be more-or-less intuitive, eg. addition should work similar to blending (yellow) + light blue) - Igreen)). "Define operation" means give description of the operation in terms of (R.G, B) codes of the color. For example, if color ) = (R, G., B.), Icolorz) = (R2.G2, B2), and Icolor,) + colorz) = |colorz) = (R.G.B). then you have to give formulas for Rs, G3, B, in terms of R.2.1.2. and B.2. R = Fr(R.G, B., R2, G2,B). G = F(R.G.B, R2.G.B.). B = F(R.G.B, R2, G2,B)