I just want to check some solutions and ask a question.

For number five, I want to disprove the statement.
For (a,b) in R^2,
||(a,b)|| = |a| + |b|  <=> sqrt (|a|^2 + |b|^2) = |a| + |b| <=> |a|^2
+ |b|^2 = (|a| + |b|)^2 .
Since (|a| + |b|)^2 = |a|^2 + |b|^2 + 2|a||b| does not equal |a|^2 + |
b|^2 unless |a||b| = 0, then there is no inner product on R^2 such
that the associated norm is given by ||(a,b)|| = |a| + |b|.

For number six, I expanded ||u+v||^2 and ||u-v||^2 in the given
equality and eventually got the equality sqrt (u^2 + v^2) = sqrt (u^2
+ v^2). Provided that I did everything right, would this verify the
equality given in the problem?

For number seven, is there any difference between (||u + iv||^2)i and
i(||u + iv||^2)?

Thanks!

P.S. Am I not allowed to do this (posting what I did for a problem)?

It would be optimal if other students would chime in with comments.  
The intention is that this was to be a discussion group, so I'll hold  
off answering for a while...  At least I won't comment on 5 and 6.

Nope.  It's simply that people like to write complex numbers in the  
form a+bi, with a and b real.  However, bi = ib.

Sure.  For what it's worth, I'm a big proponent of study groups.  I  
think that it's great when students get together and learn from each  
other's approaches to doing a problem.  In office hours, which often  
take place in 891 Evans these days, it's fine with me if people ask  
how a problem is done and other people walk to the board and explain  
how they did it.  In the office hours, I am often pressed into the  
role of the "student who has done the problem," but it's infinitely  
preferable if that role is filled by a student.

For me, email discussion has the same basic status as face-to-face  
discussion.  (But it's less fun, and the communication is usually  
slower.)

In this course, an important thing that you learn is how to write and  
read proofs.  I think that it's very important for students' education  
that they sit down by themselves and write out their homework  
solutions (even if they've seen explanations for how to do the  
homework in office hours or study groups).  I'm sure that it's  
tempting to look at someone else's written solution as you're writing  
down your own, but I do feel that you'll end up paying the price for  
this shortcut when it comes time to write the exams.

See you later,
Ken R