This collective book presents selected topics in the modern research of Hilbert space. Throughout this book, various mathematical properties of the Hilbert space and extended Hilbert space are given, accompanied by reliable solutions and exciting applications to scientific and engineering problems. It first provides some general viewpoints on convex sets, projections, and orthogonality in Hilbert spaces and then focuses on the mild solutions, the stability, and the controllability of various classes of differential equations in Hilbert spaces and applications. It also is devoted to a discussion of the extended Hilbert space, including the hypercomplex Hilbert space, the Bargmann-Hilbert space, and the enlarged Hilbert space where various mathematical and physical applications are given. A reduced Hilbert space for model Hamiltonians is also given. Together, the book presents to readers a picture of the modern theory of Hilbert space in its complexness and usefulness. The book is accessible for graduate students and could be served as a reference for scholars.