Inner Product Spce Definition Assume tht V is ector spce oer field of sclrs F in our usge this will e. Then we define inry opertor.. :V V F [once gin in our usge this will e ] so tht the following properties hold: 1. Symmetry Gien ny u V u u (1.1). Linerity in the first rgument Gien ny u1 u V nd ny sclrs u u u u (1.) 1 1 3. Positie Definiteness Gien ny u V uu 0 u u 0 if nd if u 0 (1.3) We cll this inry opertor the inner product on V nd the ector spce with such n opertor defined on it the inner product spce. Note 1. One cn esily see tht #1 nd # implies linerity in the second rgument i.e. gien ny u V nd ny sclrs 1 u u u 1 1. Note. Linerity cn e presented in two properties:. Gien ny u1 u V u u u u. 1 1. Gien ny u V nd ny sclr u u. 1
Note 3. 0 u 0 0 u 0 0 u 0 for ny ector u. In the sme mnner u0 0. Definition We cll uu for ny uv u 1 we cll unit ector. norm of u nd lel tht with u. Any ector such tht Note 4. Using properties of inner product one cn esily show tht for ny non-null ector 1 norm u is unit ector. u u V Note 5. For ny sclr nd ny ector u we he the following u clled homogeneity. u. This property is Exmple 1. 1 1 1 1 1 1 3 3 3 3 3 3 T is n inner product on 3. It is esy to show the properties. This is proly known to you s dot product. Anlogously we would define the dot product in n. The norm of ector is lso known s the mgnitude of the ector with the following c c tht is well-known formul for the distnce etween initil nd terminl points of the ector. Exmple. C for Let e the spce of ll rel lued continuous functions on the interl. Then following is the inner product on this spce
f g f x g x dx. The only property tht would require some non-triil rgument would e the only if prt of the second prt of (1.) which would follow from the continuity of the function. We shll lee this discussion for our reder s exercise. Definition For ny squre n n mtrix A we cll trce of mtrix A the sum of entries on the min digonl i.e. 11 1 1n n 1 n tr 11 nn ii i1 n1 n nn Exmple 3. Let M mn e the spce of ll m n mtrices. The following is inner product on this spce In other words n n m n m T T T A B tr A B A B i i A ij ji jiji. i1 i1 j1 i1 j1 11 1 1 n 11 1 1 n 1 n 1 n 11 11 11 1 n1 n m1 m mn m1 m mn 1 1 n n m1 m1 m m mn mn [It is esy to check the properties of definition of inner product strightforwrd.] Note 6. The inner product s defined oe is oious generliztion of dot product nd it is sometimes clled Froenius inner product due to the gret Germn mthemticin Ferdinnd Georg Froenius. 3
The following theorem is one of the importnt results nd proly the most fmous nlyticl inequlity. Theorem 1. [Cuchy-Schwrtz-Bunykosky] Gien ny u V where V is n inner product spce V the following holds. u u (1.4) with the equlity holding if nd only if u nd re linerly dependent. Proof: Assume tht is non-null ector else (1.4) holds triilly. Let t u then u u u u u u u u 0 t t t t which implies u u nd (1.4) follows. The rest of the theorem is sttement out necessry nd sufficient condition for the equlity in (1.4). We strted the proof with 0 u t u t. Oiously y positie definiteness when u t u t 0 then u t 0 which implies liner dependence etween u nd. Conersely when u nd re linerly dependent then either one of them is null ector in which cse (1.4) ecomes 0 0. If neither of them is null ector then we he u k which implies u k k k k u. Therefore we he completed the proof of the theorem. //// [Antoine-Louis Cuchy proed the inequlity oe for the dot product in 181 while Victor Bunykosky proed the inequlity for the inner product in the context of our exmple in 1859. 4
f t g t dt f t dt g t dt which is (1.4) squred. Hermn Schwrtz generlized this integrl inequlity to complex integrls in 1988. There is further generliztion of this integrl inequlity known in Rel nd Functionl Anlysis s Hölder Inequlity due to Otto Hölder nd pplicle to Leesque spces. The riety of pplictions of C-B-S inequlity in ll sorts of inner product spces is truly stonishing.] We he the following importnt theorem out the norm. Theorem. {Tringle Inequlity] Gien ny u V where V is n inner product spce V the following holds. u u (1.5) with equlity holding if nd only if the ectors re linerly dependent. Proof: This is simple consequence of theorem 1. u u u u u u u u u u u u u u nd fter tking squre root we he (1.5). If the equlity holds then liner dependence of ectors follows from preious theorem. If the ectors re linerly dependent then u u holds s well nd the equlity in (1.5) follows gin from the preious theorem. n Since the norm represents mgnitude of the ector in i.e. the distnce from the initil point to the terminl point of the ector we he cler isul explntion of the preious theorem gien in the following figure. //// 5
Figure 1. Tringle Inequlity Theorem 3. [Prllelogrm Lw] When the norm is defined in the inner product spce V y u u u then the following property holds u u u (1.6) Proof: Strightforwrd u u u u u u u u u u u u u //// [The preious theorem is one of the rillint results proly known to Apollonius of Tyn lmost 000 yers go. It generlizes Pythgors Theorem. If follows esily from known Apollonius theorem.] 6
There re norms tht re defined on ector spces tht do not come from inner products s you cn see in the following exmple. Exmple 4. Define. It is ery esy to see tht this is norm on. Howeer the 1 0 Prllelogrm Lw fils ecuse if we use u 0 1 then we he u 4 8 u u. There re other more importnt exmples of normed ectors spces tht re not inner product ectors spces most notly in Rel nd Functionl Anlysis. We shll lee this discussion outside the scope of our course. Homework: Check online. 7