This is an announcement for the paper "Banach Spaces with respect to operator-valued norms" by Yun-Su Kim. Abstract: We introduce the notions of L(H)-valued norms and Banach spaces with respect to L(H)-valued norms. In particular, we introduce Hilbert spaces with respect to L(H)-valued inner products. In addition, we provide several fundamental examples of Hilbert spaces with respect to L(H)-valued inner products. Archive classification: math.FA math.OA Mathematics Subject Classification: 46B45 ; 46C07 Remarks: 13 page The source file(s), 2o-Banach.tex: 41954 bytes, is(are) stored in gzipped form as 0803.0041.gz with size 10kb. The corresponding postcript file has gzipped size 83kb. Submitted from: kimys@indiana.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0803.0041 or http://arXiv.org/abs/0803.0041 or by email in unzipped form by transmitting an empty message with subject line uget 0803.0041 or in gzipped form by using subject line get 0803.0041 to: math@arXiv.org.