I remember loosely that I saw somewhere a space notation $C_{\to0}^0(\mathbb R^n)$ for the space of all continuous function that vanishes at infinity. But I can't find a reference. Does anyone know a textbook that use this notation?
For books which doesn't use $C_0(\mathbb R^n)$ for function vanishing at infinity, so far I only found Grafako's Classical and Modern Fourier Analysis, which use $\mathscr C^0_{00}(\mathbb R^n)$ (or $C^0_{00}(\mathbb R^n)$ if you don't care about the fontype of $C$).
I am writing this question because I am to write a notes for my teaching, where I want to give a list of function spaces for $C^0$ and $C^k$ functions.