Is There a Vector Space Defined by Canonical Boundary Conditions?

[CPL.Luke]

how do you prove/show that there really is a vector space defined by certain boundary conditions?

unfortunatly this part of pde's was glossed over in my professor's lecture notes and I don't recall him talking about it in class.

 

[CPL.Luke]

haha actually I think I just realized how to do it, you have to show that any two functions that any combination of funcions that meet the conditions must be a function that also meets the condition, which will be met with any set of boundary conditions where the function =0 at that point.

af(0) +bf'(0)=0+0=0is this correct?