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 $$\theta$$is 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