Abstract
In the present paper, the fuzzy Schwarz inequality in inner product spaces
is derived. It is an extension of the Schwarz inequality, and is described by using a
fuzzy norm and a fuzzy inner product defined by Zadeh's extension principle. The
fuzzy norm of a fuzzy set is the image of the fuzzy set under the crisp norm, and it is
also a fuzzy set. The fuzzy inner product between two fuzzy sets is the image of the
two fuzzy sets under the crisp inner product, and it is also a fuzzy set. The Schwarz
inequality evaluates the inner product between two vectors in an inner product space
by norms of the two vectors. On the other hand, the fuzzy Schwarz inequality evaluates
the fuzzy inner product between two fuzzy sets on an inner product space by fuzzy
norms of the two fuzzy sets.
