Basic Question about notation in the space of continuous functions

Question

I am reading the book "Introduction to Calculus of Variations" by Bernard Dacorogna (Could not find a link in google books) where he defines $C(\bar{\Omega})$ to be the space of continuous functions $u : \Omega \to \mathbb{R}$ which can be continuously extended to $\bar{\Omega}$ (here $\Omega$ is an open set in $\mathbb{R}^n$).

After this, he defines the norm over $C(\bar{\Omega})$ by $\|u\|_0 = \sup_{x \in \bar{\Omega}} |u(x)|$ and then $C(\bar{\Omega})$ with this norm is a Banach Space.

I am confused about two things:-

1) Does $C(\bar{\Omega})$ consist of functions $u : \Omega \to \mathbb{R}$ which can be continuously extended or functions $u : \bar{\Omega} \to \mathbb{R}$

2)The "norm" $\|.\|_0$ is not a norm as $\bar{\Omega}$ need not be bounded and hence it can take an infinite value

Can someone please let me know if I am right and if so, this notation is OK in some texts.

Answer 1

I included a screenshot below. The concept of $C(\overline{\Omega})$ is unambigious, as user161825 pointed out. If a continuous extension to the closure exists, it is unique and we may consider the function as already extended.

Item 2) is a mess-up on the author's part. Just assume $\Omega$ is a bounded open set, because it will be practically everywhere in the book.

^{I might add that this is not a book I would recommend for learning the calculus of variations. The variational chapters in Brezis and Evans would be a better source, partly because they are better written, partly because they are not laser-beam focused on the Italian tradition of the subject.}

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