Mathematical Tools for Physics - Department of Physics - University

12—Tensors 321

Exercises

1 On the three dimensional vector space of real quadratic polynomials inx, define the linear functional

F(f) =

∫ 1

0 dxf(x). Suppose that^1 , x, andx

(^2) are an orthonormal basis, then what vector A~

represents this functionalFso thatA~.~v=F(f), where the vector~vmeans a quadratic polynomial,

asf(x) =a+bx+cx^2. Ans:A~= 1 +^12 x+^13 x^2

2 In the preceding example, take the scalar product to be

〈

f,g

〉

=

∫ 1

− 1 dxf(x)g(x)and find the vector

that represents the functional in this case. Ans:A~=^12 +^34 x

3 In the first exercise above, what if the polynomials can have arbitrary order? Ans:A~=−x^1 ln(1−x)

(Not quite right because this answer is not a polynomial, so it is not a vector in the original space.
There really is no correct answer to this question as stated.)