Image-kernel orthogonality for an adjoint ($\mathrm{im}T=(\ker T^{\ast }{)}^{\perp }$)

For a linear map $T:V\to W$ between finite-dimensional inner product spaces,The first equality follows directly from the defining identity for the adjoint operator; the second follows by taking orthogonal complements. In an infinite-dimensional Hilbert space, the second equality generally requires closure of the image.