Is $\mathbb{R}^n \subset \mathbb{R}^n$? [closed]

Is it correct to say that $\mathbb{R}^n \subset \mathbb{R}^n$ ?

EDIT: The context of may question is that I am having a function that is defined as $f \colon D \subset \mathbb{R}^n \to \mathbb{R}^n $ and I am wondering if I can just generalize the definition as $f \colon \mathbb{R}^n \to \mathbb{R}^n $. The symbol $\subset$ refers to set inclusion.

^{I was just reading "J. M. Ortega, Iterative Solution of Nonlinear Equations in Several Variables". I hope I will not cause more agitation.}

Solution 1:

The most used convention is that the symbol $\subset $ applies to any subset of a set, so $A\subset A$ is correct for every set $A$.

Solution 2:

The symbol “$\subset$” could mean different things depending on the author.

In your case, I suggest trying to figure it out from the context. If there are no other restrictions on $D$ (e.g. bounded,compact,…), then it is very likely that you can also take $D=\mathbb R^n$. If you can’t figure it out in this example, try to find other occurrences in the book. Also, check if by any chance there is a section of the book dedicated to notations.

Just a small comment: in analysis I can’t recall so many statements that hold for a certain class of possibly unbounded subsets $\mathbb R^n$ (or $\mathbb C^n$) with the only exception being the whole space. Two examples that I have in mind are the Riemann mapping theorem and the strict inclusion between the Sobolev spaces $W^{1,p}(\Omega)$ and $W^{1,p}_0(\Omega)$.