Suppose $X$ is a normed space, $x_0\in X$ and $M$ is a subspace of $X$. Suppose that $d=\inf_{m\in M} \|x_0-m\|>0$ and let $W=\text{Span } M\cup\{x_0\}$. Show that the linear functional $f\colon W\to \mathbb{C}$ which vanishes on $M$ and $f(x_0)=1$ is bounded and $\|f\|=1/d$.
I have no difficulty solving this problem under the additional condition that $X$ is a Hilbert space by using an orthogonal projection of vectors on $\overline{M}$.
- Can someone give me a hint please on how to solve this problem in normed spaces?
- Since I tried to solve the general case using the Riesz Lemma unsuccessfully it would also be nice to see if a (second) solution based on this lemma and the notion of approximate orthogonality exists for the problem.