How do isomorphisms allow us to transfer properties between (isomorphic) vector spaces?

Question

Working with finite dimensional spaces is said to be easier, because every finite dimensional space over the reals of dimension $n$ is isomorphic to $\mathbb{R}^{n}$. I know what isomorphisms are and how they work, but I don't fully understand why this fact is particularly useful. I get that proving results in $\mathbb{R}^{n}$ in easier in many situations, but why do results on the reals imply something on a general vector space that is isomorphic with it?

Answer 1

My two cents: With the necessary caveat in place (that these vector spaces be real), the isomorphism preserves the basic linear algebra properties one is usually asked to prove things about in a first course.

For example, vectors $v_1$ and $v_2$ are linearly independent in the first vector space if and only if $T(v_1)$ and $T(v_2)$ are in the second. A subspace has dimension $n$ if and only if its image does too, a vector is in the kernel of an operator if and only if it is in the kernel of the right composition with your isomorphism, and etc. for whatever "vector space" property you want to think about.

When you start to put a metric on the space, you will start to ask for more form your isomorphisms. In this case, you have that all finite dimensional vector spaces are not only isomorphic, but there is a bi-Lipschitz isomorphism between them. Then, you get to transfer not just algebraic qualities from one vector space to another, but topological ones as well i.e. a sequence in the first space converges, $x_n\to x$ if and only if $Tx_n\to Tx$, etc.

I bring this up because the topological properties is where I think finite dimensional vector spaces are really "easier." I disagree that it is just because they are all isomorphic to each other. After all, every separable, infinite dimensional Hilbert space is even unitarily equivalent to $\ell^2(\mathbb{N})$, but that doesn't make them "easy," nor has that perspective been particularly useful, at least to me.

Since you have tagged real analysis, maybe I can mention Heine-Borel as a lovely feature of finite dimensional vector spaces over $\mathbb{R}$ that does simplify things quite a bit. Compactness is an important property, as extracting convergent subsequences is crucial to many arguments in real analysis. That this condition is true for so many sets and easy to check in finite dimensions is quite nice.

Furthermore, in infinite dimensional spaces, there are lots of different kinds of convergences to keep track of. In finite dimensions, they are all equivalent. Indeed, every metric you place on these spaces is equivalent (see bi-Lipschitz isomorphism)! As long as you don't care too much about how fast things converge, that means you can choose a convenient metric for your problem most of the time.

Answer 2

When you say that every finite dimensional vector space of dimension $n$ is isomorphic to $\mathbb R^n$, this is not true in general. This is only true for real vector spaces.

Now isomorphism are interesting because those allows to transfer properties from one space to another one.