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Suppose {H,(⋅,⋅)} is a complete
inner product space and H1 is a dense subspace of H. In case T is a linear
transformation from H1 to H1 (perhaps not bounded), a necessary and sufficient
condition is obtained in Theorem 1 for the existence of an inner product (⋅,⋅)1 for H1
such that (i) the identity is continuous from {H1,(⋅,⋅)1} to {H,(⋅,⋅)} and (ii) T
is bounded in {H1,(⋅,⋅)1}. When this condition holds, the inverse-shadow
inner product is defined on H1, for sufficiently large positive numbers β, by
(x,y)β,T = Σp=0∞((T∕β)px,(T∕β)py). An extension of Theorem 1 provides a
necessary and sufficient condition for the existence of an inner product (⋅,⋅)1 for H1
such that {H1,(⋅,⋅)1} is complete and (i) and (ii) hold. This latter condition, stated
in Theorem 5 in terms of a pair of inverse-shadow inner products, depends on a
description of those complete inner product spaces {H1,(⋅,⋅)1}, with H1 dense in H,
for which (i) holds. According to this description, given in Theorem 4, each
such inner product (⋅,⋅)1 is a scalar-multiple of an inverse-shadow inner
product (⋅,⋅)δ,c, where C is a bounded operator on H mapping H1 to H1 and
δ = 1.
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