Yes, the domain D you described is compact since it is the cartesian product of the compact intervals [a_i,b_i], and the continuous functions from a compact subset of R^n (or more generally from any compact hausdorff space) to R, equipped with the supremum norm, conform a Banach space.

Is it a normed space with a given norm ??

I'm not sure if I understand your question, but Banach spaces are complete normed spaces, so yes, the norm you defined is well-defined for C(D), making it a normed space (also Banach).

Yes, but it is sup norm for D=[a,b] but here D is different

D can be any compact hausdorff topological space and the result remains valid. In particular it works for the cartesian product of n compact intervals, as is your case here.

The problem is caused by the very bad formulation of the question

by Maisuria.

And, the fact that ResearcGate does not allow the use of LaTeX

commands at questions and answers .

However, to overcome this difficulty one can freely add Pdf files

containing everything in more readable form.

Pdf for a given problem. @Arpad Szaz

Dear Maisuria,

Thank for the pdf.

The formulation of its first line is

equally bad.

With best wishes,

Arpad Szaz

Let $\Omega$ be any compact subset of $R^{d}$. Then $C(\Omega)$ is always a Banach space with the sup norm. And if $\Omega$ is any open subset of $R^{d}$, then $C(\Omega)$:= the set of all continuous bounded real-valued functions, is always a Banach space with the sup norm.

Arpad Szaz

Sougata Mahanta , What is d in $R^{d}$?

Dear Maisuria,

The form of the first line is now quite good.

But, why not an arbitrary compact space is here?

In the subsequent parts, "a" and "function", and

"(for the given norm)" are superfluous.

Instead of R, you can still take a Banach space.

However, your question and my remarks are only to

have a friendly conversation.

With best regards,

Arpad Szaz

Namaste Sir Arpad Szaz

I know that (C(D),||.||) is a Banach space. (Where D is any closed and connected subset of the real line and the norm is sup-norm.)

But here, I want to know that it's true for any D, which is a compact and connected subset for $R^{n}$.

It may be a complete space because D is a compact subset of $R^{n}$. But is it a normed space?

Sir, please suggest a book for Banach space.

Thank you, Sir

Dear Maisuria,

Your latest letter again contans very false assertions.

I am not dealing with Banach spaces, but I think that

the book

An Introduction to Banach Space Theory

by R.E. Megginson

is certainly the best one.

With best regards,

Arpad Szaz

Manan Anilkumar Maisuria $R^{d}$ is the d dimensional Euclidean space, you may consider n instead of d. Finite product compact subsets of $R$ is compact. So, $[a_{1}, b_{1}]x...x[a_{d},b_{d}]$ is compact in $R^{d}$.

Sougata Mahanta Thank you