Notations:(adsbygoogle = window.adsbygoogle || []).push({});

V denotes a vector space

A, B, C, D denote subspaces of V respectively

≈ denotes the isomorphic relationship of the left and right operand

dim(?) denotes the dimension of "?"

Question:

Find a vector space V and decompositions:

V = A ⊕ B = C ⊕ D

with A≈C but B and D are not isomorphic.

My opinion:

dim(V)=dim(A)+dim(B)=dim(C)+dim(D) and dim(A)=dim(C), but dim(B)≠dim(D) since V may not be finite-dimensional. It's an idea not an example, would you make a concrete example of V?

Thanks for any help!

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# About the isomorphism of 2 infinite-dimensional vector spaces

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