**Edit:** According to the comment of Prof. Majer, I revise the question:

For a metric space $X$, we put $A=\{f:X\to \mathbb{C}\mid \text{f is bounded}\}$. We define two semi norm on $A$

$$\parallel f \parallel_{1}= \parallel \omega_{f}\parallel_{\infty}$$ or $$\parallel f \parallel_{2}= \int_{[0,\;1]} \omega_{f}(x)dx$$

where $\omega_{f}$ is the standard oscillation function..(**Edit:** In the later norm, we put $X=[0\;1]$)

We obtain two normed space on an appropriate quotient space. Are these resulting space, complete? After a possible completion how can we compare them with each other and how can we compare them with $A/C$ where $C$ is the space continuous functions and $A$ is equipped with sup norm. (By comparison I mean comparison as two Banach space). What type of topological or metric properties of $X$ is encoded in each of the above three Banach space. To what extent the endomorphism of these Banach space is classified.(Motivated by the fact that the endomorphisms of $C(X)$, as a banach space, are classified)