Given two vector spaces $X$ and $Y$ equipped with norm $|| \cdot||_X$ and $|| \cdot||_Y$, respectively. I want to know whether the following statements are equivalent.
(1). $X$ and $Y$ are isometric.
(2). There exists a isometric isomorphism between $X$ and $Y$.
(3). $X$ and $Y$ are isometrically isomorphic.
I have checked the wiki, and found that (1) is equivalent to there exists a bijective mapping $f$ which preserves the distance, i.e., $$d_Y(f(a),f(b))=d_X(a,b), \forall a,b \in X.$$
Also, (2) is equivalent to there exists a bijective linear mapping $f$ which preserves the norm, i.e., $$||f(a)||_Y=||a||_X, \forall a \in X.$$
I am not quite sure whether the equivalent statements in wiki are correct. If true, can we say (1), (2), and (3) are equivalent statements?
(2) and (3) are just different ways of saying the same things. (1) is implied by (2) since $$d(f(a),f(b))=\|f(a)-f(b)\|=\|f(a-b)\|=\|a-b\|=d(a,b)$$ (1) can be a property of metric spaces that are not necessarily vector spaces. So it is possible for normed spaces to be isometric but are not isomorphic as vector spaces. For example $\mathbb{C}$ and $\mathbb{R}^2$ are isometric (with the Euclidean distance) but it doesn't even make sense to say they are isomorphic as vector spaces since they don't have the same field.