Relationship between Complex Vector Spaces and Real Vector Spaces

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May 31st 2008, 11:00 PM
PhoenixM

Relationship between Complex Vector Spaces and Real Vector Spaces

The conclusion might be so obvious and use from time to time without proof.
How could one prove that every complex vector space is a real vector space? Is there anyway to establish a formal proof of it and ensure that it works in an infinite-dimensional case?
June 1st 2008, 06:05 AM
ThePerfectHacker

Quote:

Originally Posted by PhoenixM

The conclusion might be so obvious and use from time to time without proof.
How could one prove that every complex vector space is a real vector space? Is there anyway to establish a formal proof of it and ensure that it works in an infinite-dimensional case?

I am not exactly sure what you mean. If you are asking is $\mathbb{C}/\mathbb{R}$ a vector space, then it certainly it, because in general $E/F$ is a vector space if $E,F$ are fields with $F\subseteq E$ . The space $\mathbb{C}/\mathbb{R}$ is two dimensional with basis $\{ 1, i\}$ . You can also form $\mathbb{C}/\mathbb{Q}$ this one is infinitely-dimensional with a basis having cardinality $2^{\aleph_0}$ .
June 1st 2008, 01:46 PM
PhoenixM

I mean if X is a complex vector space, how could one prove that X is a real vector space.
June 1st 2008, 01:51 PM
ThePerfectHacker

Quote:

Originally Posted by PhoenixM

I mean if X is a complex vector space, how could one prove that X is a real vector space.

Again, what does "complex vector space" mean? But I think I understand now.

$X$ contains $\mathbb{C}$ and it satisfies the vector-space properties. Now check whether the same properties are satisfied when we think of $X$ over $\mathbb{R}$ .