On 3 Apr 2004, Dylan Thurston wrote:

You might want to add a functional dependency, if you only have one type of scalars per vertor space:

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class VectorSpace v a | v -> awhere zero :: v add :: v -> v -> v scale :: a -> v -> v

But then again, you might not.

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instance Num a => VectorSpace a a where zero = 0 add = (+) scale = (*)

I see ... functional dependency means "for each 'v' there is only one 'a' that forms a VectorSpace with 'v'". Thus the "VectorSpace a a" instance would exclude any other instance. Since this instance causes 'undecidable instance' and 'overlapping instance' errors in each constellation I tried I finally decided to content myself with special instances instance VectorSpace Double Double instance VectorSpace Float Float These will be the anchors for each type construction recursion needed for more complex VectorSpaces. But e.g. instance VectorSpace (Complex Double) (Complex Double) is still rejected as overlapping with instance (RealFloat v, VectorSpace a v) => VectorSpace a (Complex v) of course ... unfortunately. In addition I separated a class VectorSpacePure from the VectorSpace class that contains the operations that don't need the scalar type 'a', i.e. 'zero' and 'add'. Further I swapped consistently 'a' and 'v' in the class definition and in the quantity data type, i.e. "class VectorSpace a v" and "data Quantity a v". I suppose that it will more oftenly be necessary to say something about the vectors that can built from a certain scalar (i.e. Quantity a) rather than about the scalar types that can be the base of a vector type.

By the way, depending how you resolve the issue above, you might want instead

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instance (RealFloat a, VectorSpace b a) => VectorSpace [b] a where ...

That's really a nice generalization! With the specialised instances for Float and Double as shown above this works. Now I can built VectorSpace over several levels, e.g. a list of complex numbers as vectors with respect to real numbers. But it prohibits functional dependency, right? Functional dependency would probably allow the compiler to detect which scalar type is associated with an vector and thus the compiler wouldn't ask me to tell it explicitly all the time.

Alternatively, if you want to consider varying the scalars, you can add 'a' as a dummy type variable to 'Quantity':

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data Quantity v a = Quantity v

instance (Show v, Fractional a, Normed v a) => Show (Quantity v a) where show (Quantity v) = let nv::a = norm v in (show (scale (1/nv) v)) ++ "*" ++ (show nv)

Since I can't use functional dependency I chose this way. Maybe a type class like the VectorSpace finds the way to a revised version of Prelude? Thanks a lot for the help!