Linear Functionals

1. Nov 20, 2007

sharkboy

How do I solve this problem- I know it has something to do Riesz represenation but am having difficulty connecting dots

Conside R4 with usual inner product. Find the linear funcitonal associated to the vector (1,1,2,2).

What am I missing- is this problem complete or is there something else Also what does usual inner product mean.

Sharkie

2. Nov 21, 2007

morphism

Did you read the http://planetmath.org/encyclopedia/RieszRepresentationTheorem.html [Broken]? I would suspect this question is asking you to find the continuous linear functional on R^4 (a Hilbert space) associated with u=(1,1,2,2) (u is used as in the notation on the planetmath website).

Last edited by a moderator: May 3, 2017
3. Nov 21, 2007

sharkboy

So in the inner product what is x?

What is definition of inner product.

4. Nov 21, 2007

sharkboy

Wasn't clear in my last post:

there were 2 questions

1- So in the inner product (in planetmath.org) what is x?

2- What is definition of inner product.

5. Nov 21, 2007

morphism

The usual inner product on R^4 is the dot product.

6. Nov 21, 2007

HallsofIvy

Staff Emeritus
If I remember correctly, the "linear functional" associated with the given vector v in an inner product space is just f(x)= <v, x> where < , > is the inner product.

7. Dec 1, 2007

sharkboy

Morphism - Do I just <v,v> for the linear functional. I don't clearly understand what the other term I need.

Sharkie

8. Dec 1, 2007

morphism

Is f(x) = <(1,2), x> a linear functional on R^2?

9. Dec 2, 2007

sharkboy

Yes.. the way to prove is below

f(u) = <(1,2), u> = 1*u + 2*u = u + 2u = 3u
f(g) = <(1,2), g> = = 3g
f(u+g) = <(1,2), (u,g)> = 1*u + 1*g + 2*u + 2*g = 3u + 3g

f(u) + f(g) = f(u+g)

Scalar multiplication

f(kx) = <(1,2), kx> = kx*1 + 2*kx = 3kx
kf(x) = k <(1,2), x> = k(1*x + 2*x) = 3kx

Since it is closed under both scalar multiplication and additoin, it is a linear functional.

But how does it help me in my actual problem ?
Sorry I don't see the angle

Sharkie

10. Dec 2, 2007

HallsofIvy

Staff Emeritus
Your question has been answered several times in these responses! The linear functional associated with vector v is f(x)= <v, x>.

Oh, and since R4 is finite dimensional, talking about "Hilbert Spaces" and "Riesz representation" is overkill!

11. Dec 6, 2007

sharkboy

OK - I think I was missing a key info and finally figured it out. On more reading, I realised that a linear functional maps into a scalar. Thats the key I was missing. And all the other exampls made sense then. However, this problem is not

Consider the linear functional f:Rn --> R defined by f(x1,x2,. . .,xn)= x1 + x2 +. . . + xn. Find the vector u in Rn such that for all vectors v in Rn we have f(v)=(v,u), where ( , ) is the usual inner product.

The reason it isn't make sense is where does vector x come into the picture.

12. Dec 6, 2007

Office_Shredder

Staff Emeritus
They're asking what vector you have to dot with x1,...,xn to get x1+...+xn

13. Dec 6, 2007

sharkboy

Isn't that just the (1.......1) vector (size 1 x n)

14. Dec 6, 2007

Office_Shredder

Staff Emeritus
Yeah