Let V be a finite-dimensional real vector space with inner product <⋅,⋅> and L: V → R a linear transformation. Show that there exists a unique vector a ∈ V such that L(x) = <a,x>.
Hey everyone, so I'm a physics student who had to choose a few electives in the maths department. Unfortunately I picked a proof based module without realizing it as I thought it would've been a carry on from first year Linear Algebra. I decided to soldier on through as I think it might be useful down the line. I am however having difficulty unlike the maths students as they do this sort of question answering a lot. So I hope by explaining what I know someone will be able to help me to answer this question correctly.
The Attempt at a Solution
I know that the inner product for real numbers is the dot product. I know the properties the inner product satisfy (associative and commutative). Its the L(x) = <a,x> part that I don't fully understand. If anyone could help me I'd appreciate it. Thanks in advance.