北大数学物理方法(A)-数学物理方程教案12内积空间与函数空间

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Outline1›ùSȘm†¼ê˜mÔnÆ2007cSC.S.Wu1›ùSȘm†¼ê˜mOutlineùLJ:1SȘmSȆSȘm552¼ê˜m¼êSȼê8˜58˜¼ê85C.S.Wu1›ùSȘm†¼ê˜mOutlineùLJ:1SȘmSȆSȘm552¼ê˜m¼êSȼê8˜58˜¼ê85C.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceReferencesÇÂÁ§5êÆÔn{6§§18.1,18.2C.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessùLJ:1SȘmSȆSȘm552¼ê˜m¼êSȼê8˜58˜¼ê85C.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessn‘¥þ˜m½Âƒx,y,z,···(¡¥þ)8Ü¡¢(E)¥þ˜m§XJeún¤á(1)?‰˜é¥þx†y§k\{$Ž§=3éA¥þx+y§¡x†yƒÚ§äke5Ÿµ(a)x+y=y+x(b)x+(y+z)=(x+y)+z(c)3˜¥þ0§¦éuz‡x§x+0=x(d)éuz‡¥þx§3˜¥þ§P−x§¦x+(−x)=0(2)?‰¥þx9¢(E)êα§kê¦$Ž§=3éA¥þαx§¦(a)α(βx)=(αβ)x(b)(α+β)x=αx+βx(c)α(x+y)=αx+αy(d)1x=xC.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessn‘¥þ˜m½Âƒx,y,z,···(¡¥þ)8Ü¡¢(E)¥þ˜m§XJeún¤á(1)?‰˜é¥þx†y§k\{$Ž§=3éA¥þx+y§¡x†yƒÚ§äke5Ÿµ(a)x+y=y+x(b)x+(y+z)=(x+y)+z(c)3˜¥þ0§¦éuz‡x§x+0=x(d)éuz‡¥þx§3˜¥þ§P−x§¦x+(−x)=0(2)?‰¥þx9¢(E)êα§kê¦$Ž§=3éA¥þαx§¦(a)α(βx)=(αβ)x(b)(α+β)x=αx+βx(c)α(x+y)=αx+αy(d)1x=xC.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessn‘¥þ˜m½Âƒx,y,z,···(¡¥þ)8Ü¡¢(E)¥þ˜m§XJeún¤á(1)?‰˜é¥þx†y§k\{$Ž§=3éA¥þx+y§¡x†yƒÚ§äke5Ÿµ(a)x+y=y+x(b)x+(y+z)=(x+y)+z(c)3˜¥þ0§¦éuz‡x§x+0=x(d)éuz‡¥þx§3˜¥þ§P−x§¦x+(−x)=0(2)?‰¥þx9¢(E)êα§kê¦$Ž§=3éA¥þαx§¦(a)α(βx)=(αβ)x(b)(α+β)x=αx+βx(c)α(x+y)=αx+αy(d)1x=xC.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessSȌ±rn‘¥þ˜m¥¥þÝVgí2n‘¥þ˜md§k½Ân‘¥þSÈ3êKþ½Ân‘¥þ˜mV§§ƒ(¥þ)^x,y,···L«C.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessSȌ±rn‘¥þ˜m¥¥þÝVgí2n‘¥þ˜md§k½Ân‘¥þSÈ3êKþ½Ân‘¥þ˜mV§§ƒ(¥þ)^x,y,···L«C.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessSȌ±rn‘¥þ˜m¥¥þÝVgí2n‘¥þ˜md§k½Ân‘¥þSÈ3êKþ½Ân‘¥þ˜mV§§ƒ(¥þ)^x,y,···L«C.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompleteness¢n‘¥þ˜m¥Sȉ½¢n‘¥þ˜m(=K¢ê)§3À½˜|Ä{ei,i=1,2,···,n}ƒ§˜m¥?¿˜‡¥þxь±^§3ù˜|ÄþÝK(‹I)x1,x2,···,xnL«x=x1e1+x2e2+···+xnen=nXi=1xieiDefinitionéu˜m¥¥þxÚy§~„SȽÂ(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyiC.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompleteness¢n‘¥þ˜m¥Sȉ½¢n‘¥þ˜m(=K¢ê)§3À½˜|Ä{ei,i=1,2,···,n}ƒ§˜m¥?¿˜‡¥þxь±^§3ù˜|ÄþÝK(‹I)x1,x2,···,xnL«x=x1e1+x2e2+···+xnen=nXi=1xieiDefinitionéu˜m¥¥þxÚy§~„SȽÂ(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyiC.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompleteness¢n‘¥þ˜m¥SÈDefinitionéu˜m¥¥þxÚy§~„SȽÂ(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyi(x,y)´˜‡¢ê(x,y)=(y,x)(x,x)≥0…=x=0ž§âk(x,x)=0Definition:¥þxÝkxkkxk=(x,x)1/2C.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompleteness¢n‘¥þ˜m¥SÈDefinitionéu˜m¥¥þxÚy§~„SȽÂ(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyi(x,y)´˜‡¢ê(x,y)=(y,x)(x,x)≥0…=x=0ž§âk(x,x)=0Definition:¥þxÝkxkkxk=(x,x)1/2C.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompleteness¢n‘¥þ˜m¥SÈDefinitionéu˜m¥¥þxÚy§~„SȽÂ(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyi(x,y)´˜‡¢ê(x,y)=(y,x)(x,x)≥0…=x=0ž§âk(x,x)=0Definition:¥þxÝkxkkxk=(x,x)1/2C.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompleteness¢n‘¥þ˜m¥SÈDefinitionéu˜m¥¥þxÚy§~„SȽÂ(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyi(x,y)´˜‡¢ê(x,y)=(y,x)(x,x)≥0…=x=0ž§âk(x,x)=0Definition:¥þxÝkxkkxk=(x,x)1/2C.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompleteness¢n‘¥þ˜m¥SÈDefinitionéu˜m¥¥þxÚy§~„SȽÂ(x,y)=x1y1+x2y2+···+xnyn=nXi=1xiyi(x,y)´˜‡¢ê(x,y)=(y,x)(x,x)≥0…=x=0ž§âk(x,x)=0Definition:¥þxÝkxkkxk=(x,x)1/2C.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessEn‘¥þ˜m¥SÈéuEn‘¥þ˜m§XJE3þãSȽ§K¥þÝҌUØ´¢ê±¥þÝE´¢ê§7Ly¥þÚ§gSȏؔ3±ݽÂcJe§òSȽ½Â?U(x,y)=x∗1y1+x∗2y2+···+x∗nyn=nXi=1x∗iyiÙ¥x∗i´xiEÝw,§3E¥þ˜m¥(x,y)=(y,x)∗C.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessEn‘¥þ˜m¥SÈéuEn‘¥þ˜m§XJE3þãSȽ§K¥þÝҌUØ´¢ê±¥þÝE´¢ê§7Ly¥þÚ§gSȏؔ3±ݽÂcJe§òSȽ½Â?U(x,y)=x∗1y1+x∗2y2+···+x∗nyn=nXi=1x∗iyiÙ¥x∗i´xiEÝw,§3E¥þ˜m¥(x,y)=(y,x)∗C.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessEn‘¥þ˜m¥SÈéuEn‘¥þ˜m§XJE3þãSȽ§K¥þÝҌUØ´¢ê±¥þÝE´¢ê§7Ly¥þÚ§gSȏؔ3±ݽÂcJe§òSȽ½Â?U(x,y)=x∗1y1+x∗2y2+···+x∗nyn=nXi=1x∗iyiÙ¥x∗i´xiEÝw,§3E¥þ˜m¥(x,y)=(y,x)∗C.S.Wu1›ùSȘm†¼ê˜mInnerProductSpaceFunctionSpaceInnerProduct&InnerProductSpaceOrthogonalityCompletenessEn‘¥þ˜m¥SÈéuEn‘¥þ˜m§XJE3þãSȽ§K¥þÝҌUØ´¢ê±¥þÝE´¢ê§7Ly¥þÚ§gSȏؔ3±ݽÂcJe§òSȽ½Â?U(x,y)=x∗1y1+x∗2y2+···+x∗nyn=nXi=1x∗iyiÙ¥x∗i´xiEÝw,§3E¥þ˜m¥(x,

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