Question

Example of closes linear map between banach spaces which ia not open

Answer 1

Many linear equations may be formulated in terms of a suitable linear operator

acting on a Banach space. In this chapter, we study Banach spaces and linear oper-

ators acting on Banach spaces in greater detail. We give the definition of a Banach

space and illustrate it with a number of examples. We show that a linear operator

is continuous if and only if it is bounded, define the norm of a bounded linear op-

erator, and study some properties of bounded linear operators. Unbounded linear

operators are also important in applications: for example, differential operators are

typically unbounded. We will study them in later chapters, in the simpler context

of Hilbert spaces.

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