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Definition. Let V , W be normed vector spaces (both over R or over C). A linear transformation or linear operator T : V > W is bounded if there is. In this chapter we describe some important classes of bounded linear operators on. Hilbert spaces, including projections, unitary operators, and self-adjoint Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix. · Any Let X and Y be two normed linear spaces over the same field of scalars F. Definition 5.1.3: A linear operator T : X > Y is said to be continuous at x0 ? X if Our first key result related bounded operators to continuous operators. Theorem 4.3. Let V and W be normed spaces and T : V > W a linear operator. Then the 4 Linear Transformations and characterization of bounded linear functionals Let T : X ?> Y be a bounded linear transformation X,Y normed spaces, then. It is a basic fact in functional analysis that a linear operator A on a Hilbert space X is a bounded linear operator on X if and only if A is continuous at a. Its content is quite standard and this theory can be seen as a special instance of bounded linear operators on more general Banach spaces. 1.1 Hilbert space.