1) Characterize all solutions of

$$ Ax^2 + By^2 + Cz^2 + Dx + Ey + Fz + G = 0 $$

2) Corral, pages 63 and 64, all problems

3) Corral, page 70, prob. 19-21, page 74, prob. 27-28

4) Corral, page 82, prob. 19-27

5) Extra Credit, understand example 4.51 in Griffel. Present it to me in my office or some other place with a black/white board.

6) Look up the Stone-Weierstrass Theorem and understand the theorem and it's proof. You can start with Wikipedia if you like, but I really like G F Simmons "Introduction to Topology and Modern Analysis"

7) Corral, page 88, prob. 11-13

8) Corral, page 100, prob 1-5

9) Assume $f \geq 0$. I showed in class how $\int_a^b f(x)dx = \int_0^{\infty}\lambda(y) dy$, where $\lambda(y) \equiv \mu\{ x: f(x) \geq y\}$. Show how to get a similar formula for $\int_a^b f^2(x) dx$. Due Oct 14

10) Extra Credit: One way to view functional analysis is as the infinite dimensional analogue of vector calculus, except that in this case, linear transformations are already complicated enough to spend a significant amount of time on them alone. So functional analysis is infinite dimensional linear algebra and the real heart of calculus in infinite dimensional spaces moves to the non-linear parts of functional analysis, termed, naturally enough nonlinear functional analysis. This problem concerns the generalization of the notion of gradient to Hilbert spaces.

Show that the gradient of the functional $F(u) \equiv \int_{\Omega} |\nabla u|^2 dx$ is the (infinite dimensional) vector $\Delta u$. Here are some steps: (1) show that $ \frac{d}{dt}(F(u+th))|_{t=0} = \int_{\Omega} (\Delta u) h dx$ where we assume that $h(x)=0$ on $\partial \Omega$, the boundary of $\Omega$. (2) Notice that $\int_{\Omega} (\Delta u) h dx$ is a linear operator on $h$, look up the Riesz representation theorem for $L^2$ functions and say why this implies that $\nabla F = \Delta u$. This is a simple example of the Euler-Lagrange approach to variational calculus (the classical approach, in fact).