This is a question from my textbook
I'm not quite sure what C[0, 1] mean, I tried to google the similar question and found that $C[0,1]$ usually denotes the collection of continuous functions $f: [0,1]\to \mathbb{R}$, but I'm still not quite sure what $f: [0,1]$ means, does $[0,1]$ means the domain of the function, can anyone give me a straightforward example
Best Answer
$f:A\to B$ means that $f$ is a function whose domain is $A$ and such that $f(a)\in B$ for every $a\in A.$
$C[0,1]$ is the set of continuous functions from $[0,1]$ into $\mathcal R.$
$C[0,1]$ is a vector space over the reals under the definition $(r f+g)(t)=r f(t)+g(t)$ for $f,g \in C[0,1]$ and $r\in \mathcal R$ and every $t\in [0,1].$
The zero-vector of the space $C[0,1]$ is the constant function that maps $[0,1]$ onto the set $\{0\}.$ So what you are asked is: If $r_1,r_2\in \mathcal R$ and if $r_1\cos t+r_2\sin t=0$ for every $t\in [0,1]$ then does $r_1=r_2=0?$