Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2011 / Algebra / 2 / a / i / Solution

Use the convention that the inner product is linear in its first argument, as required by the displayed representation of a linear functional. Choose an orthonormal basis $e_1,\dots ,e_n$, for instance by Gram-Schmidt orthogonalization. DefineFor $u={\sum }_j{u}_j{e}_j$, conjugate linearity in the second argument givesThusThe representation is unique as well: if two vectors represent the same functional, their difference is orthogonal to every vector and therefore to itself, so it is zero.