# Spanning Sets in Inner Product Spaces

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 $x = y$.

• Hi there, welcome to Math.SE. Please take a look at the FAQ (linked at the top of every page). In particular, here's how to ask a homework question (show what you've tried, don't just copy a problem from the textbook), and here's how to format math. Apr 12 '13 at 1:14

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 $x$.
• @user72195 It does $(x,x)=0$. Now $(x,x)=\|x\|^2$. So? Apr 12 '13 at 1:46