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 conditions
However , 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
Best Answer
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.