It is proved that for every
non-zero continuous linear functional f on a non-reflexive Banach space X,
there is a closed linear subspace Y so that f∣Y does not attain its norm. In
fact, Y may be chosen with ∥f∣Y∥ arbitrarily close to ∥f∥. It is also shown
that every continuous linear functional on an infinite-dimensional normed
linear space fails to attain its norm on some linear subspace. The class of
non-zero Banach space operators which map closed bounded convex sets to
closed sets is identified as the class of Tauberian operators. (A bounded
linear operator T : X → Y is defined to be Tauberian provided T∗∗x∗∗∈ Y
implies x∗∗∈ X.) Other closed image characterizations are obtained. In
particular, using the very first result stated above, a non-zero operator is found to
be Tauberian if and only if the image of the ball of any closed subspace is
closed. The new characterizations show that the “hereditary versions” of
semi-embeddings and Fσ-embeddings are precisely the one-to-one Tauberian
operators.
