I'm doing a question on proving dimension theorem. I've thought of case 1 and I'm not sure if it is the correct approach. Please help me check it and comment it.

I have to prove using the following facts.

Given facts :

Let U be an m-dimensional subspace of and let V be a k-dimensional subspace of U, where 0<k<m

a) Any orthonormal basis for V can be expanded to form an orthonormal basis for U

b) If , then U=V\bigoplusW

My proof:

Case 1 :

Let basis for U =

By Gram-Schmidt Process, we can find an orthonormal basis for U =

Let basis for U+V =

By Gram-Schmidt Process, we can find an orthonormal basis for U+V =

U is a subspace in U+V

By part (a) in the given fact, we know that can be expanded to

Let basis for V =

By Gram-Schmidt Process, we can find an orthonormal basis for V =

dim V=m-k

dim U=k

dim (U+V) = m

and here come the result