I need to show that a linear functional $\displaystyle \alpha_v$ on a Hilbert space $\displaystyle H$ defined by $\displaystyle \alpha_v(w)=<w,v>$ has operator norm $\displaystyle \|\alpha_v\|_{op}=\|v\|$.

I have already shown by the Cauchy Schwarz inequality that $\displaystyle \|\alpha_v\|_{op}\leq\|v\|$.

I think now I need to show that $\displaystyle \|\alpha_v\|_{op}\geq\|v\|$.

Any help would be great, Thanks