Spanning Sets in Inner Product Spaces

Question

Let $v_1, v_2,\ldots, v_n$ 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 $(x, v_k) = 0$ for every $k$, then $x = 0$.

c) If $(x, v_k) = (y, v_k)$ for every $k$, then $x = y$.

Answer 1

a) One appropriate choice of $v$ suffices. Try $v=x$.

b) Write $x=\lambda_1v_1+\ldots+\lambda_nv_n$ and use the bilinearity of the inner product when computing $(x,x)=(x,\lambda_1v_1+\ldots+\lambda_nv_n)$.

c) Apply b) to $x-y$ instead fo $x$.