The largest part of Chap. 4 is classical. It is concerned with the concepts of closed and closable semibouned below quadratic forms. We discuss the first and second representation theorems, as well as the bijective correspondence between closed semibounded below forms and the associated selfadjoint operators. In Sect. 4.3 we introduce the so-called energy space of a semibounded symmetric (in particular, selfadjoint) operator A and discuss the Friedrichs construction of its extension \(\widehat A_{\mathrm {F}}\) preserving the lower bound of A. Section 4.5 presents form sums of a semibounded selfadjoint operator and a form, the KLMN theorem and its applications, as well as some new facts on the Bessel operator on the half-line. The construction of the Kreı̆n extension and extremal properties of the Friedrichs and Kreı̆n extensions are exhibited in Sects. 4.6 and 4.7. All results are illustrated by means of numerous examples of ordinary and partial differential operators. We also discuss in Sects. 4.9 Kato’s lower semicontinuity criterion for a form to be closed as well as his result on monotone form convergence from below. This result is then applied for proving the strong resolvent convergence of von Neumann extensions.