I was reading a functional analysis book and one of the corollaries of Hahn-Banach Theorem in there is
$\begin{array}{l}\text { Corollary } 19.1 .5 \text { (Hahn-Banach). Let } X \text { be a normed linear space. Suppose that: } \\ \text { (a) } M \text { is a closed subspace of } X, \\ \text { (b) } x_{0} \in X \backslash M, \text { and } \\ \text { (c) } d=\operatorname{dist}\left(x_{0}, M\right)=\inf \left\{\left\|x_{0}-m\right\|: m \in M\right\} \\ \text { Then there exists a functional } \mu \in X^{*} \text { such that } \\ \qquad \mu\left(x_{0}\right)=1,\left.\quad \mu\right|_{M}=0, \quad \text { and } \quad\|\mu\|=\frac{1}{d}\end{array}$
This is different from what I get from other books which states $\mu\left(x_{0}\right)=d,\left.\quad \mu\right|_{M}=0, \quad \text { and } \quad\|\mu\|=1$.
They look a lot similar, are they the same?
Best Answer
$\mu \in X^{*}$ iff $d\mu \in X^{*}$ so thee are the same. (If $\mu$ is as in the second statement change it to $\frac 1 d \mu$ to get the first one; similarly the other way).