How do you determine a subspace of C01SolutionTo show that a

How do you determine a subspace of C[0,1]?

Solution

To show that a space is a subspace of another space we need to have defined two spaces.
You have defined S but what space are we to show it is a subspace of?
I\'m guessing that X = {f: ƒ C[0,1]}, i.e. the set of all continuous functions on [0,1]. The problem is to show S is a subspace of X.
Go to the definition of subspace. Unfortunately there are different definitions of subspaces, but the one I\'m guessing you are using is:
A set B is a subspace of a set C if:
1. B is contained in C
2. B is closed under addition.
3. B is closed under scalar multiplication.

All three of these properties following directly from the definition of S and the theorems about integration. You need to know your definitions to do math proofs.