prove that vector spaces are isomorphic if and only if they have the same dimension

A,B,C,D,E please

The axioms for a vector space V can be used to prove the elementary properties for a vector space. Because of Axiom 2, Axioms 2 and 4 imply, respectively, that 0 + u= u and -u+u = 0 for all u. Complete the proof that -u is unique by showing that if u + w=0, then w=-u. Use the ten axioms of a vector space to justify each step. Axioms In the following axioms, u, v, and w are in vector space V and c and d are scalars. 1. The sum u + v is in V. 2. u+v=V+u 3. (u+v)+w=u+(V+W) 4. V has a vector 0 such that u + 0 = u. 5. For each u in V, there is a vector - u in V such that u + (-1)u = 0. 6. The scalar multiple cu is in V. 7. c(u + v) = cu + cv 8. (c + d)ucu + du 9. c(du) = (cd)u 10. 1u=u Suppose that w satisfies u+w=0. Adding - u to both sides results in the following. (-u) + [u+w] =(-u) +0 [(-u) +u]+w=(-u) +0 by Axiom 2

Show that if X and Y Hausdorff spaces, then so is the product space X × Y. are