Solution space of linear homogeneous PDE forms a vector space?

[kingwinner]

Homework Statement

Claim:
The solution space of a linear homogeneous PDE Lu=0 (where L is a linear operator) forms a "vector space".

Proof:
Assume Lu=0 and Lv=0 (i.e. have two solutions)
(i) By linearity, L(u+v)=Lu+Lv=0
(ii) By linearity, L(au)=a(Lu)=(a)(0)=0
=> any linear combination of the solutions of a linear homoegenous PDE solves the PDE
=> it forms a vector space

Homework Equations

N/A

3. The Attempt at a Solution and comments
Now, I don't understand why ONLY by proving (i) and (ii) alone would lead us to conclude that it is a vector space. There are like TEN properties that we have to prove before we can say that it is a vector sapce, am I not right?
Are there any theorem or alternative definition that they have been using?

Thanks!

 

[lanedance]

maybe consider the other axioms and test them, some may have been considered more bovious, but always worth checking

 

[Office_Shredder]

Other things are satisfied by the definition of functions, for example u+v=v+u is obvious and not a property of being a solution of a PDE. It may be implicitly assumed that you're talking about the subset of the vector space of all functions from U to V (where U and V are R^m and Rⁿ such that you'd be looking for solutions with that domain and codomain). In that case you only have to prove the two subspace properties