Exploration of indispensable Banach-space valued functions

In the paper, we present a necessary and sufficient condition for the existence of a sequence of measurable functions with finite values, which converge to any given essential bounded function in the topology of essential supremum in a Banach space. A new convergence method is proposed, which allows for the discovery of an essential bounded function $F$ that is valued in a Banach space. Generally speaking, there exists a Banach-valued essential bounded function $F$ which $F_n$ can't converge to $F$ in the topology of essential supremum for any sequence of finite-valued measurable function.