 Dual Spaces, Transposes and AdjointsBalmohan V. Limaye ()
Chapter 4 in Linear Functional Analysis for Scientists and Engineers, 2016, pp 119-158 from Springer Abstract: Abstract 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. Date: 2016 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-10-0972-3_4 Ordering information: This item can be ordered from DOI: 10.1007/978-981-10-0972-3_4 Access Statistics for this chapter More chapters in Springer Books from Springer |
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