Let's consider the category of finite dimensional real vector spaces (VS) / inner product spaces (IPS).
Which of the following pairs of isomorphic vector spaces (given appropriate dim constraints), are naturally isomorphic as VS/IPS? Are the natural isomorphisms unique? In the case IPS, is there always a natural isomorphism that is also isometric?
| Domain | Co-Domain | nat. iso. as VS? | nat. iso. as IPS? | nat. iso. unqiue? |
|---|---|---|---|---|
| $V$ | $V$ | ✔ | ✔ | ? |
| $V$ | $V^*$ | ✘ | ✔? | ? |
| $V$ | $(V^*)^*$ | ✔ | ✔ | ? |
| $U⊕V$ | $V⊕U$ | ✔ | ✔ | ? |
| $U⊗V$ | $V⊗U$ | ✔ | ✔ | ? |
| $U⊕(V⊕W)$ | $(U⊕V)⊕W$ | ✔ | ✔ | ? |
| $U⊗(V⊕W)$ | $(U⊗V)⊗W$ | ✔ | ✔ | ? |
| $U⊗(V⊕W)$ | $(U⊗V)⊕(U⊗W)$ | ✔ | ✔ | ? |
| $(U,V)$ | $U^*⊗V$ | ✔ | ✔ | ? |
| $(U⊗V, W)$ | $(U, (V,W))$ | ✔ | ✔ | ? |
| $ℝ^m⊕ℝ^n$ | $ℝ^{m+n}$ | if $m+n≤1$? | if $m+n≤1$? | ? |
| $ℝ^m⊗ℝ^n$ | $ℝ^{m⋅n}$ | if $m⋅n∈\{0,1,m,n\}$? | if $m⋅n∈\{0,1,m,n\}$? | ? |
| $ℝ^m⊕ℝ^n$ | $ℝ^k⊕ℝ^l$ | if $\{k,l\}=\{m,n\}$? | if $\{k,l\}=\{m,n\}$? | ? |
| $ℝ^m⊗ℝ^n$ | $ℝ^k⊗ℝ^l$ | if $\{k,l\}=\{m,n\}$? | if $\{k,l\}=\{m,n\}$? | ? |
| $ℝ^m⊗ℝ^n$ | $ℝ^n⊕…⊕ℝ^n$ | if $m≤1$? | if $m≤1$? | ? |
Feel free to suggest other common isomorphic pairs / edit the list.