TITLE: Prove that the set of all real valued function on [a,b] is a vector space
QUESTION [8 upvotes]: Question - Show that the set of all real valued functions on [a,b] , $\mathrm F $[a,b] under usual addition and scalar multiplication is a vector space. 
What I did - let  $\mathrm L = \{ F:[a,b] \; | \; a,b \in \mathbb R \} \; $ & $\; \mathrm u, v \in L$ s.t  $\mathrm u = f[a,b] \; \& \;  v= g[a,b] $and after that I showed that the 10 axioms do satisfy under these conditionsHowever , my instructor has marked it all wrong and she has highlighted that I have started the problem in the wrong way. She also have added the correct method , which is as follows ;  
  $\mathrm L = \{ F:[a,b]\rightarrow \mathbb R^{2} \; | \; a,b \in \mathbb R \} \; $  let  $\; \mathrm x,y \in [a,b]$ Take any$ \,\mathrm F_1$ & $ \, \mathrm F_2$ in $\mathrm L \, $ s.t  $\mathrm F_1(x,y) \in \mathbb R^{2} \; \& \;  F_2(x,y) \in \mathbb R^{2} $ Now Im really confused  because:  1.what is wrong with my approach ? , what is my mistake ? 2.why my instructor have used functions defined on (x,y) ? I think it should be [a,b][please look into this photo to see if I have missed something else

REPLY [3 votes]: It seems your instructor has erred and written that  $L$ consists of functions from  $\Bbb R^2$to $\Bbb R^2$, or else  (possibly) this was the original problem.  We can't see the original problem so who knows?
For any vector space $V$ and set $S$, the set of functions from $S$ to $V$, denoted $V^S$, is a vector space in a natural way.