I want to know more about $BV[0,1]$. Like the way a function in $BV[0,1]$ acts on $C[0,1]$, and when a sequence in $C[0,1]$ is weakly convergence?
2026-03-29 18:32:39.1774809159
What is the dual space of $C[0,1]$?
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A sequence $f_n \in C[0,1]$ is weakly convergent to $f\in C[0,1]$ if and only if for any $g\in BV[0,1]$ we have $$\int_0^1 (f_n (u) - f(u) )dg(u) \to 0 $$ as $n\to \infty.$
Where the above integral is a $Riemman-Stielties $ integral.