Determine whether a set is subspace or not

[mithila]

The set of all functions f such that the integral of f(x) with respect to x over the interval [a,b] is
1. equal to zero
2. not equal to zero
3. equal to one
4. greater than equal to one
etc.

How can we determine this types of set is a subspace of not.



for the case of the set of continuous function we can easily check the closure conditions but for integrable function how can we check that.

can anyone help me to solve this type of problem

Thanks.

 

[CompuChip]

Usually, you just check the conditions. For example, take two functions f and g whose integral over [a, b] is zero, and check that the integral of (f + g) over [a, b] is also zero, etc.