I am wondering if anything is known about the dual space of the space of all functions $f: [0,1]\to R$, that have finite one-sided limits in every point, endowed with $sup$-norm? More exactly, let $M$ be the space of functions satisfying $f(x)=1/2 (f (x+0)+f(x-0))$ at any $x\in (0,1)$ and $f(0+)=f(0)$, $f(1-)=f(1)$ with $sup$-norm. Is there any description for $M*$?
Or, maybe, for the space of piecewise continuous functions on $[0,1]$?
> I am wondering if anything is known about the dual space of the > space of all functions $f: [0,1]\to R$, that have finite one-sided > limits in every point, endowed with $sup$-norm? > More exactly, let $M$ be the space of functions satisfying $f(x)=1/2 > (f > (x+0)+f(x-0))$ at any $x\in (0,1)$ and $f(0+)=f(0)$, $f(1-)=f(1)$ with > $sup$-norm. Is there any description for $M*$?
Consider this space to be C(K), where K is the two-arrows space. (top and bottom of the lexicographic square, Sorgenfrey interval). Since K is compact Hausdorff, the dual is M(K), the space of signed measures on K.
> Or, maybe, for the space of piecewise continuous functions on $[0,1]$?
> > I am wondering if anything is known about the dual space of the > > space of all functions $f: [0,1]\to R$, that have finite one-sided > > limits in every point, endowed with $sup$-norm? > > More exactly, let $M$ be the space of functions satisfying $f(x)=1/2 > > (f > > (x+0)+f(x-0))$ at any $x\in (0,1)$ and $f(0+)=f(0)$, $f(1-)=f(1)$ with > > $sup$-norm. Is there any description for $M*$?
> Consider this space to be C(K), where K is the two-arrows space. > (top and bottom of the lexicographic square, Sorgenfrey interval). > Since K is compact Hausdorff, the dual is M(K), the space of > signed measures on K.
> > Or, maybe, for the space of piecewise continuous functions on $[0,1]$?
