Mathematical Tools for Physics

6—Vector Spaces 154

9 Scalar products and norms used here are just like those used for example 5. The difference is that the sums

go from 1 to infinity. The problem of convergence doesn’t occur because there are only a finite number of

non-zero terms.

10 Take the norm to be

‖(a 1 ,a 2 ,...)‖=

√∑

∞

k=1

|ak|^2 , (9)

and this by assumption will converge. The natural scalar product is like that of example 5, but with the sum

going out to infinity. It requires a small amount of proof to show that this will converge. See problem 19.

11 A norm is‖~v‖=

∑∞

i=1|ai|. There is no scalar product that will produce this norm, a fact that you can

prove by using the results of problem 13.

13 A natural norm is

‖f‖=

[∫

b

a

dx|f(x)|p

] 1 /p

. (10)

To demonstrate that thisisa norm requires the use of some special inequalities found in advanced calculus

books.

15 If AandB are two matrices, a scalar product is

〈

A,B

〉

= Tr(A†B), where†is the transpose complex

conjugate of the matrix andTrmeans the trace, the sum of the diagonal elements. Several possible norms

can occur. One is‖A‖=

√

Tr(A†A). Another is the maximum value of‖A~u‖, where~uis a unit vector and

the norm of~uis taken to be

[

|u 1 |^2 +···+|un|^2

] 1 / 2

.

19 A valid definition of a norm for the motions of a drumhead is its total energy, kinetic plus potential. How do
you describe this mathematically? It’s something like

∫

dxdy

1

2

[(

∂f

∂t

) 2

+

(

∇f

) 2

]

I’ve left out all the necessary constants, such as mass density of the drumhead and tension in the drumhead.

You can perhaps use dimensional analysis to surmise where they go.