let me take a funtion that vanish at end points x=0 and x=L. do these functions qualify to be a vector space?
i proceeded this way - to show that they qualify, i have to show that they are closed, commutative etc. i think they are not closed. so they dont form vector space.am i correct??
how about periodic functions obeying f(0)=f(L)?(hints please)
and functions that obey f(0)=5?(hints please)
many thanks.

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Assuming that you mean, literally, all functions such that f(0)= 0 and f(L)= 0 for some specific number, L, then suppose f(0)= 0, f(L)= 0, g(0)= 0, and g(L)= 0. What is (f+g)(0)? What is (f+g)(L)?

If f(0)= 0 and f(L)= 0, what is (af)(0) and (af)(L) for any number a?
You do not need to show "commutativity", "distributive law" etc. because you already know those are true of numbers and your operations on functions are defined by (f+ g)(x)= f(x)+ g(x), and (af)(x)= a(f(x)), in terms of operations on numbers.