VectorSpace

trait VectorSpace[A] { } // isomorphisms 
trait Iso[A, B] { 
  def to(a: A): B 
  def from(b: B): A 
} 
trait TensorAlgebra { 
  type * [V] 
  type ⨂ [V, W] 
  type ⨁ [V, W] 
  type Z
  type I

  implicit def dual[V](implicit V: VectorSpace[V]): VectorSpace[*[V]] 
  implicit def sum[V, W](implicit V: VectorSpace[V], W: VectorSpace[W]): VectorSpace[V ⨁ W] 
  implicit def product[V, W](implicit V: VectorSpace[V], W: VectorSpace[W]): VectorSpace[V ⨂ W] 
  
  // add implicit VectorSpace's everywhere 
  
  def zero[V]: Iso[V, V ⨁ Z] 
  def unit[V]: Iso[V, V ⨂ I] 
  def commuteSum[V, W]: Iso[V ⨁ W, W ⨁ V] 
  def commuteProd[V, W]: Iso[V ⨂ W, W ⨂ V] 
  def directSum[V, W](v: V, w: W): V ⨁ W 
  def inner[V, U, W](v: V ⨂ *[U], w: U ⨂ W): V ⨂ W 
  def outerProduct[V, W](v: V, w: W): V ⨂ W 
  def differentiate[V, W](f: V => W): W ⨂ *[V] 
} 

trait Module { 
  val T: TensorAlgebra 
  import T._ 
}