Mathematical Tools for Physics

6—Vector Spaces 162

6.11 Which of these are vector spaces?
(a) all polynomials of degree 3
(b) all polynomials of degree≤ 3 [Is there a difference between (a) and (b)?]
(c) all functions such thatf(1) = 2f(2)
(d) all functions such thatf(2) =f(1) + 1
(e) all functions satisfyingf(x+ 2π) =f(x)
(f) all positive functions

(g) all polynomials of degree≤ 4 satisfying

∫ 1

− 1 dxxf(x) = 0.

6.12 For the common picture of arrows in three dimensions, prove that the subset of vectors~v that satisfy
A~.~v= 0for fixedA~forms a vector space. Sketch it.
(b) What if the requirement is that bothA~.~v= 0andB~.~v= 0hold. Describe this and sketch it.

6.13 If a norm is defined in terms of a scalar product,‖~v‖=

√〈

~v,~v

〉

, then it satisfies the “polarization identity”

(for real scalars),
‖~u+~v‖^2 +‖~u−~v‖^2 = 2‖~u‖^2 + 2‖~v‖^2. (22)

6.14 If a norm satisfies the polarization identity, then it comes from a scalar product. Again, assume real scalars.
[Consider combinations of‖~u+~v‖^2 , ‖~u−~v‖^2 and construct the scalar product.]

6.15 Modify the example number 2 of section6.3so thatf 3 =f 1 +f 2 meansf 3 (x) =f 1 (x−a) +f 2 (x−b)
for fixedaandb. Is this still a vector space?

6.16 The scalar product you use depends on the problem you’re solving. The fundamental equation (5.12)
started from the equationu′′=λuand resulted in the scalar product

〈

u 2 ,u 1

〉

=

∫b

a

dxu 2 (x)*u 1 (x)

Start instead from the equationu′′=λw(x)uand see what identity like that of Eq. (5.12) you come to. Assume
wis real.