in finite dimensions using matrix transformations Consider the special case of (where is an integer) with the standard inner product where are represented as column matrices and with respect to the standard orthonormal basis on (here, is at its th coordinate and everywhere else; as usual, will now be associated with the dual basis) and where denotes the conjugate transpose of Let be any linear functional and let be the unique scalars such that where it can be shown that for all Then the Riesz representation of is the vector To see why, identify every vector in with the column matrix so that is identified with As usual, also identify the linear functional with its transformation matrix, which is the row matrix so that and the function is the assignment where the right hand side is matrix multiplication. Then for all which shows that satisfies the defining condition of the Riesz representation of The bijective antilinear isometry defined in the corollary to the Riesz representation theorem is the assignment that sends to the linear functional on defined by where under the identification of vectors in with column matrices and vector in with row matrices, is just the assignment As described in the corollary, 's inverse is the antilinear isometry which was just shown above to be: where in terms of matrices, is the assignment Thus in terms of matrices, each of and is just the operation of conjugate transposition (although between different spaces of matrices: if is identified with the space of all column (respectively, row) matrices then is identified with the space of all row (respectively, column) matrices). This example used the standard inner product, which is the map but if a different inner product is used, such as where is any Hermitian positive-definite matrix, or if a different orthonormal basis is used then the transformation matrices, and thus also the above formulas, will be different. Relationship with the associated real Hilbert space Assume that is a complex Hilbert space with inner product When the Hilbert space is reinterpreted as a real Hilbert space then it will be denoted by where the (real) inner-product on is the real part of 's inner product; that is: The norm on induced by is equal to the original norm on and the continuous dual space of is the set of all -valued bounded -linear functionals on (see the article about the polarization