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For the space of continuous and bounded functions $\mathbb{R} \to \mathbb{R}$, we can always define the supremum norm without any problems.

I say that a function $f: \mathbb{R} \to \mathbb{C}$ is bounded if the function $|f|: \mathbb{R} \to \mathbb{R}$ is bounded in the standard sense. (I don't know if this is standard terminology. Also note that $| \cdot|$ denotes the modulus.)

Questions:

1. Can we use the supremum norm on bounded continuous functions $\mathbb{R} \to \mathbb{R}$ to define a norm on the space of bounded continuous functions $\mathbb{R} \to \mathbb{C}$?

2. If so, can we take this even further and find a norm on the space of bounded functions $\mathbb{R} \to \mathbb{C}$ continuous at the origin (but not necessarily anywhere else)?

Motivation: By Bochner's theorem, the characteristic function $f$ of any finite measure on $\mathbb{R}$ with the Borel $\sigma$-algebra inherited from the Euclidean topology is continuous at the origin and bounded by $f(0)$. So if such a norm as asked for in 2. exists, it would provide another way to define a topology on the space of such finite measures, due to the one to one correspondence between finite measures on $\mathbb{R}$ and their characteristic functions. (See also)

Chill2Macht
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    It seems to me that this should be easy. The space $\mathcal B$ of bounded functions $\mathbb R\to \mathbb C$ is Banach with the sup norm. Functions which are continuous at the origin are a closed vector subspace of $\mathcal B$, and so they too form a Banach space. – Giuseppe Negro Jun 05 '17 at 08:53
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    all norms are continuous because they are Lipschitz. – Masacroso Jun 05 '17 at 08:56
  • @GiuseppeNegro So we don't need to assume (uniform) continuity everywhere for that to work? Or am I confusing supremum with maximum (attained supremum)? I thought we might have needed uniform continuity because $\mathbb{R}$ isn't compact so we can't use Heine-Cantor. EDIT: never mind it seems https://math.stackexchange.com/questions/282121/is-the-space-of-bounded-functions-with-the-maximums-norm-a-banach-space-and-even https://math.stackexchange.com/questions/282609/is-the-space-of-bounded-functions-with-the-supremum-norm-a-banach-algebra should I delete this question? It seems dumb(er) now. – Chill2Macht Jun 05 '17 at 13:28
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    I don't think you should erase the question. The answer might be easy, but not "dumb". Moreover, the remarks at the end are interesting, and indeed, I think that something like that is done somewhere in probability. – Giuseppe Negro Jun 05 '17 at 16:49

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