Let $M$ be a closed linear subspace of a normed linear space N, and let $x_0$ be a vector not in M. If d is the distance from $x_0$ to M, show that there exists a functional $f_0$ in $N^*$ such that $f_0(M) = 0, f_0(x_0) = 1$ and $||f_0|| = 1/d$ .
I know that I can find a support functional at $x_0$ but how to proceed further?
Define a linear subspace $M_1$ by $M_1 = \mathbb{R}x_0$ + M. Note that every vector in $x \in M_1$ can be written uniquely in the form $x = \lambda x_0 + y$ where $\lambda \in \mathbb{R}$ and $y \in M$. Now define a linear functional $f_1$ on $M_1$ by $f_1(\lambda x_0 + y ) = \lambda$. By construction we have $\ker(f_1) = M$, $f_1(x_0) = 1$, and $f_1^{-1}(1) = x_0 + M.$ From this we get that the norm of $f_1$ is, $$\|f_1\|_{M_1^*} = \frac{1}{d(0, x_0 + M)} = \frac{1}{d(x_0, M)} = \frac{1}{d}.$$ Now use Hahn-Banach Thm to extend $f_1$ to $f_0$ on the entire space $X$ with norm $\|f_0\|_{X^*} = \frac{1}{d(x_0, M)} = \frac{1}{d}.$