let $\Omega$ be (locally) compact Hausdorff space.
I Denote by $C(\Omega)$ normed space of all continuous (compactly supported) functions with sup norm $$\| f \| = \sup_{x \in \Omega} f(x) $$ amd by $\mathfrak{M}(\Omega)$ space of finite (or regular) signed measures .
When from Riesz–Markov–Kakutani representation theorem it follows that where is a linear isomorphism $I : \mathfrak{M}(\Omega) \to C^*(\Omega)$ defined by the rule $$ I(\mu)(f) = \int_\Omega f \, \mathrm{d}\mu. $$ I know that in case $\Omega \subset \mathbb{R}$ I know that (local) bounded variation norm on $\mathfrak{M}(\Omega)$ will make $I$ into isometry of normed spaces. The obvious norm for general case would be push-forward $$ \| \mu \| = \| I(\mu) \| $$
But aren't there a more meaningful norm on $\mathfrak{M}(\Omega)$ with the same property?
Why total variation $\| \mu \| = |\mu |(\Omega)$ norm won't work?
Secondly it's well known that signed measure can be represented as $$ \mu = \mu^+ - \mu^-, $$ where $\mu^+$ and $\mu^-$ are normal measures. When, as $I$ maps measures into positive functionals by her structure, every functional $F$ can be written for some signed measure $\mu$ as $$ F = I(\mu) = I(\mu^+) - I(\mu^-) = F^+ - F^-, $$
where $F^+ $ and $F^{-}$ are positive functionals.
What is the way to define $(F^+,F^-)$ just in terms of $F$ without any reference to $I$?
Basicly, I want to be able to migrate this definition to other functional spaces which are not exactly $C(\Omega).$
It would be nice to know some direct references concerning topic of positive-negative decomposition of functionals too. As it feels like quiet a standard topic. (I Just want to know what part of general functional analysis texts speaks about it is generally presented, because I haven't seen it yet).
Thanks for your time and attention.
--Addendum
The construction of $F^+$ for abstract normed space is obvious if positive cone $V_+$ spans $V$. As positive cone $V_+$ spans $V$ there exists a basis $\{e_a | a \in A \}$ of exclusively positive vectors for some (uncountable) set $A$ (This needs axiom of choice). Then every vector $v$ can be written as $$ v = \sum_{a \in A}v_ae_a $$ with only finite number of non-zero $v_a$ .Then put $$ F^+(v) = \sum_{a \in A} v_a (f(e_a))^+ $$ and $$F^-(v) = \sum_{a \in A} v_a (f(e_a))^- $$ From properties of Cones it follows that for every $v \in V$ coordinates $v_a > 0$ so this functionals are indeed positive. However, It seems that this functions are bounded functionals only if norm in $V$ is invariant to a change of signs of coordinates.
This properties hold for any finite-dimensional space $\mathbb{R}^n$ trivialy and for every inner product space which admits Schauder basis of positive elements. So if $H$ is such inner product space and where exists an embedding $E:V \to H$ which preserves positivity of an element the proposed construction will work ( extend pull-back of $F E^{-1}$ from $\mathrm{im} E$ to $H$ by Hahn-Banach theorem as $G$ . Then define $F^+ = G^+ E$ and $F^{-} = G^{-} E$ ).
For $V = C(\Omega)$ role of $H$ can be played by Hilbert space $L_2(\Omega,\mu)$ where $\mu$ is a finite measure which is non-zero on every open set (Existence of such measure is another question).
This construction is bad for two reasons:
Firstly, it depends on "coordinates".
Secondly, It still relies on some external space when $V$ is not simple enough.