Let $K$ be a metric space. Denote $\mathcal{B_1}(K)$ as the class of real-valued bounded first Baire functions on $K$. A real-valued function $f:K \rightarrow \mathbb{R}$ is the first Baire function if there exists a sequence of continuous functions that converges pointwise to $f.$
Let
$D(K) = \{ f \in \mathcal{B_1}(K): f = u - v, \text{ where } u,v \geq 0 \text{ are bounded and lower semicontinuous functions.} \}$
Define a norm $\| f \|_D = \inf \{ \| u + v \|_{\infty}: f = u -v \text{ for }u, v \geq 0, \text{ bounded and lower semicontinuous functions.} \}$, where $\| f \|_{\infty} = \sup_{x \in K} f(x)$
Question: How to prove that $(D(K), \| \cdot\|_D)$ is a Banach space?
My attempt: I want to show that if $\sum \| f_n \|_D$ converges, then $\sum f_n$ converges. Hence $(D(K), \| \cdot \|_D$ is a Banach space.
Suppose $\sum \|f_n\|_D$ converges. By definition of $\| \cdot \|_D,$ there exists $u_n$ and $v_n$ such that $\sum \|f_n\|_D = \sum \| u_n + v_n \|_{\infty}$. Then I stuck here. Any hint on how to proceed would be appreciated.