MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C82F7D.6FF89A40" This document is a Single File Web Page, also known as a Web Archive file. If you are seeing this message, your browser or editor doesn't support Web Archive files. Please download a browser that supports Web Archive, such as Microsoft Internet Explorer. ------=_NextPart_01C82F7D.6FF89A40 Content-Location: file:///C:/1F38C741/main2008A.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii"

Official site of the course: www.cs.bgu.ac.il/~adv= an081

Office hours (311/58):      Sunday 18-19

   &nb= sp;            =             &nb= sp;         Tuesday 15-16

 

Definition of the inner product space (from Wikipedia= http://en.wikiped= ia.org/wiki/Inner_product_space):

Let V be a vector space over a field F (which is the real or the complex numbers)

&m= iddot;      =             &nb= sp; Inner product= is a map

3D"

Satisfying

1.<= span style=3D'font:7.0pt "Times New Roman"'>      = Conjugate symmetry:

3D"\langle

This condition implies = that 3D", because 3D"\langle

2.<= span style=3D'font:7.0pt "Times New Roman"'>      = Linearity in the first variable:

3D"\langle

3D"\langle

3.<= span style=3D'font:7.0pt "Times New Roman"'>      = Nonnegativity<= span lang=3DEN style=3D'font-size:12.5pt;font-family:Arial;color:black;mso-ansi-= language: EN'>:

3D"\langle

4.<= span style=3D'font:7.0pt "Times New Roman"'>      = Nondegeneracy<= span lang=3DEN style=3D'font-size:12.5pt;font-family:Arial;color:black;mso-ansi-= language: EN'>:

3D"\langle = implies x =3D 0.

 

<= span lang=3DEN style=3D'font-size:12.5pt;font-family:Symbol;mso-fareast-font-fam= ily: Symbol;mso-bidi-font-family:Symbol;color:black;mso-ansi-language:EN'>·            = ;        Equicontinuous convergence of a series of functions in L2[a,b] is convergence in the sup norm.

·            = ;        Inner product sp= aces have a naturally defined norm

3D"

 

<= span lang=3DEN style=3D'font-size:12.5pt;font-family:Symbol;mso-fareast-font-fam= ily: Symbol;mso-bidi-font-family:Symbol;color:black;mso-ansi-language:EN'>·            = ;        (See http://en.= wikipedia.org/wiki/Hilbert_projection_theorem):

The Hilbert Projection Theorem is a famous result of convex analysis that says that for every point x in a Hilbert space H and every closed subspace 3D"M, there exists a unique point 3D"mfor which 3D"\lVertis minimized over M. A necessary and sufficient condition for m is that the vector x − m be orthogonal to M.

 

&m= iddot;      =             &nb= sp; Here are example= s of convergences of series for functions: x, x^2, step on [-π, π]. And the Mathematica file, which performs these calculations.

 

Andrey's<= /span> hompage

BGU

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