The following is an exercise from the third edition of Functional Analysis by B V Limaye (p. 438):
Let $f$ and $g$ be two continuous linear functionals on a Hilbert space $H$. $\|g\|=\|f\|$ and $g(x)=f(x)$ for some nonzero $x$ in $\ker(f)^\perp$. Prove that $f=g$.
We used the fact that $H$ can be written as a direct sum of $\ker(f)$ and its orthogonal complement. But it's not leading to anything. If someone can help us proceed it would be great.
Wlog $f(v)=\langle x,v\rangle$ for all $v\in H.$
Let $y\in H$ s.t. $g(v)=\langle y,v\rangle$ for all $v\in H.$
By hypothesis, $\|y\|=\|x\|$ and $\langle y,x\rangle=\langle x,x\rangle$ hence by the case of equality in Cauchy-Schwarz inequality, $y=x.$