We also introduce two notions which will be needed for a general definition of the derivative of a function between vector spaces. A norm is a distance function on a vector space which is compatible with vector addition and scalar multiplication. If the vector space is complete with that norm, we call it a Banach space .
Important examples are with the p -norms, and most of the commonly used function spaces.
Let and denote the norms on defined in class. The spaces are Banach spaces for , that is, are Banach spaces, that is, for each p , the above equation defines a norm, and is complete. Thus the two norms are not equivalent. Prove that this defines a norm, and that Y is complete with that norm.