Dual Spaces, Transposes and Adjoints

In this chapter we develop a duality between a normed space X and the space $$X'$$ consisting of all bounded linear functionals on X, known as the dual space of X. As a consequence of the Hahn–Banach extension theorem, we show that $$X'\ne \{0\}$$ if $$X\ne \{0\}$$ . We also prove a companion result which is geometric in nature and is known as the Hahn–Banach separation theorem. We characterize duals of several well-known normed spaces. To a bounded linear map F from a normed space X to a normed space Y, we associate a bounded linear map $$F'$$ from $$Y'$$ to $$X'$$ , known as the transpose of F. To a bounded linear map A from a Hilbert space H to a Hilbert space G, we associate a bounded linear map $$A^*$$ from G to H, known as the adjoint of A. We study maps that are ‘well behaved’ with respect to the adjoint operation. We also introduce the numerical range of a bounded linear map from a nonzero inner product space to itself. These considerations will be useful in studying the spectral theory in the next chapter.