Outline1ùSÈm¼êmÔnÆ2007cSC.S.Wu1ùSÈm¼êmOutlineùÇ:1SÈmSÈSÈm552¼êm¼êSȼê858¼ê85C.S.Wu1ùSÈm¼êmOutlineùÇ:1SÈmSÈSÈm552¼êm¼êSȼê858¼ê85C.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceReferencesÇÂÁ§5êÆÔn{6§§18.1,18.2C.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessùÇ:1SÈmSÈSÈm552¼êm¼êSȼê858¼ê85C.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessn¥þm½Âx,y,z,···(¡¥þ)8Ü¡¢(E)¥þm§XJeún¤á(1)?é¥þxy§k\{$§=3éA¥þx+y§¡xyÚ§äke5µ(a)x+y=y+x(b)x+(y+z)=(x+y)+z(c)3¥þ0§¦éuzx§x+0=x(d)éuz¥þx§3¥þ§P−x§¦x+(−x)=0(2)?¥þx9¢(E)êα§kê¦$§=3éA¥þαx§¦(a)α(βx)=(αβ)x(b)(α+β)x=αx+βx(c)α(x+y)=αx+αy(d)1x=xC.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessn¥þm½Âx,y,z,···(¡¥þ)8Ü¡¢(E)¥þm§XJeún¤á(1)?é¥þxy§k\{$§=3éA¥þx+y§¡xyÚ§äke5µ(a)x+y=y+x(b)x+(y+z)=(x+y)+z(c)3¥þ0§¦éuzx§x+0=x(d)éuz¥þx§3¥þ§P−x§¦x+(−x)=0(2)?¥þx9¢(E)êα§kê¦$§=3éA¥þαx§¦(a)α(βx)=(αβ)x(b)(α+β)x=αx+βx(c)α(x+y)=αx+αy(d)1x=xC.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessn¥þm½Âx,y,z,···(¡¥þ)8Ü¡¢(E)¥þm§XJeún¤á(1)?é¥þxy§k\{$§=3éA¥þx+y§¡xyÚ§äke5µ(a)x+y=y+x(b)x+(y+z)=(x+y)+z(c)3¥þ0§¦éuzx§x+0=x(d)éuz¥þx§3¥þ§P−x§¦x+(−x)=0(2)?¥þx9¢(E)êα§kê¦$§=3éA¥þαx§¦(a)α(βx)=(αβ)x(b)(α+β)x=αx+βx(c)α(x+y)=αx+αy(d)1x=xC.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessSȱrn¥þm¥¥þÝVgí2n¥þmd§k½Ân¥þSÈ3êKþ½Ân¥þmV§§(¥þ)^x,y,···L«C.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessSȱrn¥þm¥¥þÝVgí2n¥þmd§k½Ân¥þSÈ3êKþ½Ân¥þmV§§(¥þ)^x,y,···L«C.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessSȱrn¥þm¥¥þÝVgí2n¥þmd§k½Ân¥þSÈ3êKþ½Ân¥þmV§§(¥þ)^x,y,···L«C.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompleteness¢n¥þm¥SȽ¢n¥þm(=K¢ê)§3À½|Ä{ei,i=1,2,···,n} §m¥?¿¥þxѱ^§3ù|ÄþÝK(I)x1,x2,···,xnL«x=x1e1+x2e2+···+xnen=nXi=1xieiDefinitionéum¥¥þxÚy§~SȽÂ(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyiC.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompleteness¢n¥þm¥SȽ¢n¥þm(=K¢ê)§3À½|Ä{ei,i=1,2,···,n} §m¥?¿¥þxѱ^§3ù|ÄþÝK(I)x1,x2,···,xnL«x=x1e1+x2e2+···+xnen=nXi=1xieiDefinitionéum¥¥þxÚy§~SȽÂ(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyiC.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompleteness¢n¥þm¥SÈDefinitionéum¥¥þxÚy§~SȽÂ(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyi(x,y)´¢ê(x,y)=(y,x)(x,x)≥0 =x=0§âk(x,x)=0Definition:¥þxÝkxkkxk=(x,x)1/2C.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompleteness¢n¥þm¥SÈDefinitionéum¥¥þxÚy§~SȽÂ(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyi(x,y)´¢ê(x,y)=(y,x)(x,x)≥0 =x=0§âk(x,x)=0Definition:¥þxÝkxkkxk=(x,x)1/2C.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompleteness¢n¥þm¥SÈDefinitionéum¥¥þxÚy§~SȽÂ(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyi(x,y)´¢ê(x,y)=(y,x)(x,x)≥0 =x=0§âk(x,x)=0Definition:¥þxÝkxkkxk=(x,x)1/2C.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompleteness¢n¥þm¥SÈDefinitionéum¥¥þxÚy§~SȽÂ(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyi(x,y)´¢ê(x,y)=(y,x)(x,x)≥0 =x=0§âk(x,x)=0Definition:¥þxÝkxkkxk=(x,x)1/2C.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompleteness¢n¥þm¥SÈDefinitionéum¥¥þxÚy§~SȽÂ(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyi(x,y)´¢ê(x,y)=(y,x)(x,x)≥0 =x=0§âk(x,x)=0Definition:¥þxÝkxkkxk=(x,x)1/2C.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessEn¥þm¥SÈéuEn¥þm§XJE3þãSȽ§K¥þÝÒUØ´¢ê±¥þÝE´¢ê§7Ly¥þÚ§gSÈØ3±Ý½ÂcJe§òSȽ½Â?U(x,y)=x∗1y1+x∗2y2+···+x∗nyn=nXi=1x∗iyiÙ¥x∗i´xiEÝw,§3E¥þm¥(x,y)=(y,x)∗C.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessEn¥þm¥SÈéuEn¥þm§XJE3þãSȽ§K¥þÝÒUØ´¢ê±¥þÝE´¢ê§7Ly¥þÚ§gSÈØ3±Ý½ÂcJe§òSȽ½Â?U(x,y)=x∗1y1+x∗2y2+···+x∗nyn=nXi=1x∗iyiÙ¥x∗i´xiEÝw,§3E¥þm¥(x,y)=(y,x)∗C.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessEn¥þm¥SÈéuEn¥þm§XJE3þãSȽ§K¥þÝÒUØ´¢ê±¥þÝE´¢ê§7Ly¥þÚ§gSÈØ3±Ý½ÂcJe§òSȽ½Â?U(x,y)=x∗1y1+x∗2y2+···+x∗nyn=nXi=1x∗iyiÙ¥x∗i´xiEÝw,§3E¥þm¥(x,y)=(y,x)∗C.S.Wu1ùSÈm¼êmInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessEn¥þm¥SÈéuEn¥þm§XJE3þãSȽ§K¥þÝÒUØ´¢ê±¥þÝE´¢ê§7Ly¥þÚ§gSÈØ3±Ý½ÂcJe§òSȽ½Â?U(x,y)=x∗1y1+x∗2y2+···+x∗nyn=nXi=1x∗iyiÙ¥x∗i´xiEÝw,§3E¥þm¥(x,