Inner Product and Linear Transformation

  • Thread starter J-Wang
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1. The problem statement, all variables and given/known data
Let V be a finite-dimensional real inner product space with inner product < , >.

Let L:V->R be a linear map.

Show that there exists a vector u in V such that L(x) = <x,u> for all x in V.


2. The attempt at a solution
It seems really simple but I just can't phrase it properly.

I know since L is linear, I can express as L(x+y) = L(x) + L(y)

I tried to make x and u be expressed as combination of orthogonal basis, and try to work in the fact that inner product will result in ZERO since they are orthogonal.

Any help would be greatly appreciated.
 

HallsofIvy

Science Advisor
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Since V is a finite dimensional vector space, there exist an orthonormal basis [itex]\{v_i\}[/itex] for i from 1 to n. Let [itex]a_i= L(v_i)[/itex]. What can you say about <x, v> with [itex]v= \sum_{i=1}^n a_iv_i[/itex]?
 
Does it has something to do with the Projection of x on the basis? :D
 

hunt_mat

Homework Helper
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Are you trying to prove the Reisz-representation theorem?

I would see how this linear functional acted on individual basis elements.
 

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