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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Proof of Wang's conjecture on subspaces of an inner product space


Authors: Dragomir Z. Dokovic and Jason Sanmiya
Journal: Proc. Amer. Math. Soc. 129 (2001), 1573-1580
MSC (1991): Primary 15A03, 15A63; Secondary 14C17, 15A42
Published electronically: February 2, 2001
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Abstract | References | Similar Articles | Additional Information

Abstract:

B.Y. Wang conjectured that if $R_{t}$ and $S_{t}$ $(t=1,\ldots ,k)$ are subspaces of an $n$-dimensional complex inner product space $V$, and their dimensions are $i_{t}$ and $n-i_{t}+1$, respectively, where $1\le i_{1}<i_{2}<\cdots <i_{k}\le n$, then there exists a $k$-dimensional subspace $W$ having two orthonormal bases $\{x_{1},\ldots ,x_{k}\}$ and $\{y_{1},\ldots ,y_{k}\}$ with $x_{t}\in R_{t}$ and $y_{t}\in S_{t}$ for all $t$.

We prove this conjecture and its real counterpart. The proof is in essence an application of a real version of the Bézout Theorem for the product of several projective spaces.


References [Enhancements On Off] (What's this?)

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Additional Information

Dragomir Z. Dokovic
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: djokovic@uwaterloo.ca

Jason Sanmiya
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: jssanmiy@uwaterloo.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06105-6
PII: S 0002-9939(01)06105-6
Keywords: Sequences of subspaces, orthonormal bases, B\'{e}zout theorem, orientation of a graph
Received by editor(s): July 30, 1999
Published electronically: February 2, 2001
Additional Notes: Supported in part by the NSERC Grant A-5285.
Communicated by: Lance W. Small
Article copyright: © Copyright 2001 American Mathematical Society