Chemical Engineering Hand Written_Notes_Part_93

Chemical Engineering Hand Written_Notes_Part_93 - 188 5....

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
188 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES 5.4. Projection Theorem in a general Hilbert Space. Equations we have derived in the above sub-sections are special cases of a very general result called projection theorem, which holds in any Hilbert space. Although we state this result here without giving a formal proof, the discussion in the above subsections provided su cient basis for understanding the theorem. Theorem 13 . Classical Projection Theorem : Let X be a Hilbert space and S be a f nite dimensional subspace of X. Corresponding to any vector y X, there is unique vector p S such that k y p k 2 k y s k 2 for any vector s S. Furthermore, a necessary and su cient condition for p S be the unique minimizing vector is that vector ( y p ) is orthogonal to S. Thus, given any f nite dimensional sub-space S spanned by linearly inde- pendent vectors © a (1) , a (2) , ....... , a ( m ) ª and an arbitrary vector y X we seek a vector p S p
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

Page1 / 2

Chemical Engineering Hand Written_Notes_Part_93 - 188 5....

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online