# For any two almost complex structures on infinite-dimensional space: Do they give isomorphic vector spaces? Are they similar?

I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much as possible, but I have been stuck on the concepts of almost complex structures and complexification. I have studied several books and articles on the matter including ones by Keith Conrad, Jordan Bell, Gregory W. Moore, Steven Roman, Suetin, Kostrikin and Mainin, Gauthier

I have several questions on the concepts of almost complex structures and complexification. Here are some:

Assumptions: Let $$W = (W,\text{Add}_W: W^2 \to W,s_W: \mathbb R \times W \to W)$$ be an $$\mathbb R$$-vector space, which may be infinite-dimensional. Suppose $$W$$ has an almost complex structure $$H \in Aut_{\mathbb R}(W)$$. $$H$$ uniquely corresponds to the $$\mathbb C$$-vector space $$(W,H)$$, where scalar multiplication is given by the complex structure $$s_W^{H}: \mathbb C \times W \to W$$, $$s_W^{H}(a+ib,v) := s_W(a,v) + s_W(b,H(v))$$. Note that $$s_W^{H}$$ agrees with the original real scalar multiplication $$s_W$$.

What I know:

• A1. For $$W$$ finite-dimensional and for any other almost complex structure $$J \in Aut_{\mathbb R}(W)$$, we have that $$(W,H)$$ and $$(W,J)$$ are $$\mathbb C$$-isomorphic but not necessarily by the identity map $$id_W$$ on $$W$$.

• A2. For $$W$$ finite-dimensional and for any other almost complex structure $$J \in Aut_{\mathbb R}(W)$$, we have that $$H$$ and $$J$$ are similar, i.e. $$H \circ S = S \circ J$$, for some $$S \in Aut_{\mathbb R}(W)$$

• A3. (A2) may be proven using (A1). See here and here. (I'm not sure if Moore (Section 9.1) uses (A1)).

Questions:

1. Is (A1) true if $$W$$ were instead infinite-dimensional? (You may use axiom of choice.)

2. Is (A2) true if $$W$$ were instead infinite-dimensional? (You may use axiom of choice.)

What I've tried:

• If yes to Question 1, then I think yes to Question 2 because I think we can still make the same argument as in the answers in the links in (A3). If no to Question 1, I think we don't necessarily have no to Question 2.
• I am puzzled how you can have a proof for this in the finite-dimensional case that does not trivially generalize to the infinite-dimensional case. – Eric Wofsey Feb 28 at 16:59
• @EricWofsey Thanks Eric Wofsey! As I edited post to include allowing axiom of choice, I forgot about simply considering bases – John Smith Kyon Mar 2 at 3:09

Yes, and I don't know of any proof in the finite-dimensional case that does not also work in the infinite-dimensional case with trivial modifications. Given an almost complex structure $$J$$ on $$W$$, you can think of $$W$$ as a $$\mathbb{C}$$-vector space via $$J$$. Pick a basis $$B$$ for this $$\mathbb{C}$$-vector space. Then $$B\cup J(B)$$ is a basis for $$W$$ as an $$\mathbb{R}$$-vector space. So, $$\dim_\mathbb{R} W=2\cdot \dim_\mathbb{C} W$$. In particular, this means that $$\dim_\mathbb{C} W$$ is uniquely determined by $$\dim_\mathbb{R} W$$ (it is half the dimension when it is finite, or the same dimension when it is infinite). So any other almost complex structure $$H$$ makes $$W$$ a $$\mathbb{C}$$-vector space of the same dimension as $$J$$ does, so they are isomorphic.
Note that question 2 is equivalent to question 1. Indeed, $$S:W\to W$$ witnesses that $$J$$ and $$H$$ are similar iff $$S$$ is an isomorphism $$(W,J)\to (W,H)$$ of complex vector spaces (this is trivial from the definitions).