1. The integral is absolutely convergent and the Petersson inner product is a positive definite Hermitian form.

2. So, an inner product on a real vector space is a positive-definite symmetric bilinear form.

3. Course Name (Linear Algebra with Applications ) Code 136.235 Description 136.235 Linear Algebra with Applications 6 credit hours Vector spaces, linear transformations, inner product spaces, eigenvalues and eigenvectors, orthogonal and Hermitian matrices, and applications.

4. If V and W are finite dimensional inner product spaces and T: V → W is a linear map, then the Adjoint T∗ is the linear transformation T∗: W → V satisfying for all v ∈ V,w ∈ W, hT(v),wi = hv,T∗(w)i