Is R^n x R^m Isomorphic to R^{n+m}?

[yifli]

Here is how I prove it:

Let \theta_1(resp.,\theta_2) be the injection from R^m(resp.,R^n) to R^{m+n}

Since injection is an isomorphism, \theta_1+\theta_2 is the isomorphism

Is this correct?

 

[gb7nash]

When you say isomorphic, do you mean does there exist a bijection between the sets? Or are you talking about ring or group isomorphism? I ask this because an isomorphism between sets is a group theory topic, but you posted in the calculus section.

 

[micromass]

And what exactly do you mean with \theta_1+\theta_2?

 

[yifli]

I mean bijection between the sets.

 

[yifli]

sum of two functions. (f+g)(x) = f(x)+g(x)

 

[yifli]

Here is how I prove it:

Let \theta_1(resp.,\theta_2) be the injection from R^m(resp.,R^n) to R^{m+n}

Since injection is an isomorphism, \theta_1+\theta_2 is the isomorphism

Is this correct?

To be more clear, injection \thetais a linear mapping from V_i to \prod{V_i} such that \theta(\alpha_j)=(0,...0,\alpha_j , 0,... ,0)

 

[vigvig]

Any 2 finite dimensional commutative vector spaces are naturally isomorphic with each other. A natural isomorphism is obtained by considering the map that sends the basis of one to the other