Mathematical Tools for Physics

6—Vector Spaces 153

there is an upper bound (many in fact) and we take the smallest of these as the norm. On−∞< x <+∞,

the function|tan−^1 x|hasπ/ 2 for its least upper bound, though it never equals that number.

5 We can take as a scalar product

〈

(a 1 ,...,an),(b 1 ,...,bn)

〉

=

∑n

k=1

a*kbk. (6)

We can just as easily have another scalar product for the same vector space, for example

〈

(a 1 ,...,an),(b 1 ,...,bn)

〉

=

∑n

k=1

k a*kbk

In fact any other positive function can appear as the coefficient in the sum and it still defines a valid scalar

product. It’s surprising how often something like this happens in real situations. In studying normal modes

of oscillation the masses of different particles will appear as coefficients in a natural scalar product.

I used complex conjugation on the first factor here, but example 5 referred to real numbers only. The reason

for leaving the conjugation in place is that when you jump to example 14 you want to allow for complex

numbers, and its harmless to put it in for the real case because in that instance it leaves the number alone.

For a norm, there are many possibilities:

‖(a 1 ,...,an)‖ 1 =

√

∑n

k=1

|ak|^2

‖(a 1 ,...,an)‖ 2 =

∑n

k=1

|ak|

‖(a 1 ,...,an)‖ 3 = maxnk=1|ak|

‖(a 1 ,...,an)‖ 4 = maxnk=1k|ak|.

(7)

The United States Postal Service prefers the second of these norms, see problem8.45.

6 A possible choice for a scalar product is

〈

f,g

〉

=

∫b

a

dxf(x)*g(x). (8)