One of my homework questions is:
Let $v_1, v_2,\dots ,v_n$ be a spanning set (in particular, a basis) in an inner product space $V$ . Prove that
a) If$(x,v)=0$ for all $v\in V$, then $x=0$;
b) If$(x,v_k)=0$ $\forall k$, then $x=0$;
c) If $(x,v_k) = (y,v_k)$ $\forall k$, then $x=y$.
I initially thought for part a) to just say do a proof where $v$ is set to $x$, which would ultimately get $(x,x)=0$, so $x$ would also have to be equal to $0$. But upon thinking about it, would that proof work since it is not specified that $x$ is in $V$, or can you still go about it that way? and if not, what step should I take first?
I am very lost on how to start part b, but was thinking if I am set on part a then b should follow more naturally. Does anyone have a recommendation for a starting point?
And for c, what I thought to do was just go by the definitions that $(x,vk)$ is the same as $||x||||v_k||$, and so we then have $||x||||v_k||=||y||||v_k||$. Can I then cancel out the $||v_k||$ to get $||x||=||y||$? Even this would not be super clean because it ends with $\pm x= \pm y$. Any suggestions how to clear this up/ get on the right track?
Sorry this is so long and detailed. Just trying to understand as thoroughly as possible.