1 00:00:17,539 --> 00:00:28,989 so let us begin with this is a quick review of axioms for norm 2 00:00:28,989 --> 00:00:43,659 so given the vector 3 00:00:43,659 --> 00:00:55,850 space x and of course the field f we wrote three axioms for defining the norm on this 4 00:00:55,850 --> 00:01:13,040 so this function called norm was a function that was from x to r plus r plus is real positive 5 00:01:13,040 --> 00:01:20,600 real numbers including zero so r plus a set of positive real numbers including zero so 6 00:01:20,600 --> 00:01:27,220 this is a function from an element in x which is the compact way of writing for element 7 00:01:27,220 --> 00:01:44,159 in x to r plus and we said the three axioms one is that norm x is greater than 0 for all 8 00:01:44,159 --> 00:01:54,880 x that belong to x and x not equal to 0 vector 9 00:01:54,880 --> 00:02:03,950 and norm x equal to 0 if and only if x equal to 0 vector so this was the first axiom the 10 00:02:03,950 --> 00:02:16,730 second axiom was norm of alpha into x is norm x mod alpha or absolute value of alpha into 11 00:02:16,730 --> 00:02:33,450 norm x where alpha is a scalar belonging to field f and a third axiom is triangle in equality 12 00:02:33,450 --> 00:02:50,269 so this states that distance of x plus y take any two elements from vector space x distance 13 00:02:50,269 --> 00:02:55,849 of the vector x plus y is always less than or equal to or the length of vector norm of 14 00:02:55,849 --> 00:03:01,419 vector of x plus y is always less than or equal to norm x plus norm y 15 00:03:01,419 --> 00:03:09,269 this is generalization of triangle equality that you know for one dimension or in three 16 00:03:09,269 --> 00:03:22,109 dimensions for triangles generalized to any other space so we said any function that satisfies 17 00:03:22,109 --> 00:03:28,389 these criteria it should be a real positive function it should give you a real number 18 00:03:28,389 --> 00:03:35,329 it should be a nonzero real number or when x is not 0 it should be 0 real when x is not 19 00:03:35,329 --> 00:03:42,629 equal to 0 vector and so on and then we saw couple of examples that are functions that 20 00:03:42,629 --> 00:03:46,310 can be classified as a norm or that cannot be classified as a norm 21 00:03:46,310 --> 00:03:53,079 so both are important because you understand something better when you see where these 22 00:03:53,079 --> 00:04:00,049 one of these axiom fails so there are multiple ways of defining norms not a unique way a 23 00:04:00,049 --> 00:04:05,370 pair of a vector space together with or a linear space together with a definition of 24 00:04:05,370 --> 00:04:14,389 norm gives you a norm vector space so that is take home message well why did we do this 25 00:04:14,389 --> 00:04:20,639 i yesterday said that we are doing all this because you know want to talk about point 26 00:04:20,639 --> 00:04:31,910 about limits and sequences so why do i need to talk about limits when we work in numerical 27 00:04:31,910 --> 00:04:39,300 methods we are forced to look at sequences of vector i just give you a very brief example 28 00:04:39,300 --> 00:04:50,680 we will actually do this much more in detail later let see i want to solve this equation 29 00:04:50,680 --> 00:04:58,630 these two are coupled equations and i want to solve them simultaneously i want to solve 30 00:04:58,630 --> 00:05:04,070 them simultaneously these kinds of problems i am writing it in an abstract form very often 31 00:05:04,070 --> 00:05:08,690 we encounter these kind of problems well i am going to write this in abstract way as 32 00:05:08,690 --> 00:05:24,850 f function 1 xy equal to 0 and function 2 xy equal to 0 there are two functions f1 xy 33 00:05:24,850 --> 00:05:34,730 equal to 0 f2 xy equal to 0 and this kind of equations arise steady state of a cstr 34 00:05:34,730 --> 00:05:37,310 concentration and temperature are linked 35 00:05:37,310 --> 00:05:41,560 so first equation could be energy balance second could be material balance and then 36 00:05:41,560 --> 00:05:47,820 you get two equations into one concentration temperature let us solve them for i am going 37 00:05:47,820 --> 00:06:00,460 to define a vector this is my function vector i am going to call this as f of x of well 38 00:06:00,460 --> 00:06:12,210 let me call some new variable equal to 0 so my eta is a vector which comprises of x and 39 00:06:12,210 --> 00:06:21,650 y and then i want to solve for f eta equal to 0 vector 40 00:06:21,650 --> 00:06:34,030 this is my 0 vector i am just writing the same thing in a different format now what 41 00:06:34,030 --> 00:06:41,060 method you know for solving this how do you solve this “professor - student conversation 42 00:06:41,060 --> 00:06:53,500 starts” bijection method pardon me bjection method bijection method is for difficult to 43 00:06:53,500 --> 00:06:58,090 scale to two variables one variable well defined bijection method is there “professor - student 44 00:06:58,090 --> 00:07:02,410 conversation ends” you can have bisection method for two variables but well let us take 45 00:07:02,410 --> 00:07:03,730 a very simple iterative scheme 46 00:07:03,730 --> 00:07:14,900 let us construct a very simple iterative scheme i will write eta plus f eta i will add this 47 00:07:14,900 --> 00:07:20,430 vector eta on both sides and then i construct an iteration whether it will convergent or 48 00:07:20,430 --> 00:07:25,990 it is a different story but i will construct an iterative process so i will start with 49 00:07:25,990 --> 00:07:33,080 some gas vector eta 0 that is let say well i do not know what the solution is so i am 50 00:07:33,080 --> 00:07:38,310 going to guess some solution so let say i start with say minus 1 and 1 51 00:07:38,310 --> 00:07:51,340 this is x this is y and then what i want to do is to say that eta k plus 1 equal to eta 52 00:07:51,340 --> 00:08:09,960 k plus f of eta k is it okay i have just formulated an iteration scheme in which i start with 53 00:08:09,960 --> 00:08:19,720 vector 0 i take the 0 vector substitute here i will get vector 1 i take vector 1 substitute 54 00:08:19,720 --> 00:08:27,250 here i will get vector 2 “professor - student conversation starts” how do i know whether 55 00:08:27,250 --> 00:08:36,419 this sequence of vectors is converging to something pardon me the difference between 56 00:08:36,419 --> 00:08:38,120 eta k plus 1 what is difference 57 00:08:38,120 --> 00:08:43,029 see it is a two dimensional vector now i will just further convenience it into two dimensional 58 00:08:43,029 --> 00:08:48,190 vector i could have done this in n dimensions i could have written this in an n dimensions 59 00:08:48,190 --> 00:08:54,029 n equations in n unknowns very very common chemical engineering starting trying to solve 60 00:08:54,029 --> 00:08:57,620 steady state energy material balance for a plant you can get 1000 equations and 1000 61 00:08:57,620 --> 00:09:04,709 unknowns okay the difference vector pardon me the difference vector but what of difference 62 00:09:04,709 --> 00:09:11,939 vector norm of the difference vector so we have to talk about a vector converging to 63 00:09:11,939 --> 00:09:17,490 another vector a vector converging to a solution what should happen at the solution 64 00:09:17,490 --> 00:09:30,100 let us say if x star is a solution eta star is a solution 65 00:09:30,100 --> 00:09:38,360 f eta equal to 0 what she say is correct that one thing is that you know should be equal 66 00:09:38,360 --> 00:09:51,870 to 0 of course at the solution so at eta start so that is f eta star equal to 0 fine but 67 00:09:51,870 --> 00:09:59,459 i am starting an iterative process so what i am going to get is i am going to get this 68 00:09:59,459 --> 00:10:11,519 vector sequence x eta 0 eta 1 2 and so on and i am going to get this vector the question 69 00:10:11,519 --> 00:10:16,820 is is the sequence is the sequence converging to eta star 70 00:10:16,820 --> 00:10:26,980 does this goes to eta star that is the question i need to answer see this is the solution 71 00:10:26,980 --> 00:10:33,221 if i had pluck eta k here it is not going to be equal to zero it is not going to be 72 00:10:33,221 --> 00:10:39,410 equal to zero so it is going to some other small number probably is it small so how do 73 00:10:39,410 --> 00:10:46,850 you answer this question in general n dimensional spaces or function spaces that is where we 74 00:10:46,850 --> 00:10:53,869 need to now talk about i may have scenario where i have a sequence of functions i have 75 00:10:53,869 --> 00:10:59,579 sequence of function and i will give an example i am going to show you a small demo also sequence 76 00:10:59,579 --> 00:11:00,579 of functions 77 00:11:00,579 --> 00:11:07,470 “professor - student conversation ends” so the question is is this sequence convergent 78 00:11:07,470 --> 00:11:14,819 so this kind of problems are always encountered in numerical analysis because almost every 79 00:11:14,819 --> 00:11:22,250 method that you have for solving you know most of the problems through computing is 80 00:11:22,250 --> 00:11:33,920 iterative you start with the guess and you come up with a new guess and so on so there 81 00:11:33,920 --> 00:11:40,839 is this need to look at convergence of sequences so we are going to define two notions one 82 00:11:40,839 --> 00:11:46,720 is cauchy sequence 83 00:11:46,720 --> 00:12:01,139 so i am taking a set of infinite set of sequences 84 00:12:01,139 --> 00:12:07,230 or infinite set of vectors which are generated by some process you know it could be some 85 00:12:07,230 --> 00:12:16,040 iterative scheme by which you are working or whatever it is now i want to know how do 86 00:12:16,040 --> 00:12:37,180 i formally define convergence a sequence of vectors is said to cauchy if difference 87 00:12:37,180 --> 00:12:44,019 between xn minus that is nth element in the sequence and m filament in the sequence if 88 00:12:44,019 --> 00:12:53,639 this tends to 0 norm of this tends to zero as m and n become infinitive so more and more 89 00:12:53,639 --> 00:12:59,199 elements are generating this the vectors come closer and closer 90 00:12:59,199 --> 00:13:12,759 well in one dimensional vector space so in one dimensional vector space that is a set 91 00:13:12,759 --> 00:13:26,399 of real numbers well when a sequence is cauchy it convergence to a limit inside a set but 92 00:13:26,399 --> 00:13:34,670 depends upon the space funny things can happen if the space is not complete what is this 93 00:13:34,670 --> 00:13:40,499 business of completeness we will come to that soon before that let me define convergence 94 00:13:40,499 --> 00:13:45,040 sequence so there are two different notions one is cauchy sequence other is convergence 95 00:13:45,040 --> 00:13:54,980 sequence these just for the sake of nice mathematics these are very very relevant to computing 96 00:13:54,980 --> 00:13:59,429 what is the convergence sequence 97 00:13:59,429 --> 00:14:06,720 so i a m considering this sequence again in fact this is a short hand notation for sequence 98 00:14:06,720 --> 00:14:12,670 i am not going to write every time k going from 0 to infinity or k going from 0 to n 99 00:14:12,670 --> 00:14:23,440 whatever it is curly braces x superscript k is a sequence in a norm linear space or 100 00:14:23,440 --> 00:14:42,820 a norm vector space now this is said to be convergent to a vector x star if this is said 101 00:14:42,820 --> 00:14:51,990 to be convergent to an element x star if difference between x star and x k goes to zero difference 102 00:14:51,990 --> 00:14:56,879 between x star and x k goes to 0 as k goes to infinity 103 00:14:56,879 --> 00:15:04,209 so what i want to show you is that it is not obvious that a cauchy sequence will always 104 00:15:04,209 --> 00:15:10,339 be convergent it depends upon the space that you are considering a convergence sequence 105 00:15:10,339 --> 00:15:17,829 is always a cauchy sequence but vice versa is not necessarily true a cauchy sequence 106 00:15:17,829 --> 00:15:25,410 may not be convergent a convergent sequence is always a cauchy sequence 107 00:15:25,410 --> 00:15:30,879 now examples will make it clear why i am talking of this funny things and we will also realize 108 00:15:30,879 --> 00:15:38,179 that this is something that you deal with every day when you use computers so i am going 109 00:15:38,179 --> 00:15:51,290 to take a example of a vector space in which a cauchy sequence is not convergent i am going 110 00:15:51,290 --> 00:15:59,309 to take a example of a vector space in which a cauchy sequence is not convergent 111 00:15:59,309 --> 00:16:09,060 so basically i want to give an example of this idea that convergence to a particular 112 00:16:09,060 --> 00:16:16,649 element is something different when it depends upon the space my first example here is my 113 00:16:16,649 --> 00:16:27,869 space x is my first example here is a set of rational numbers q and i am taking field 114 00:16:27,869 --> 00:16:36,309 f also to be q i am taking a field also to be q so this combination will form a vector 115 00:16:36,309 --> 00:16:44,439 space and i can find very easily a sequence in this vector space which is cauchy but not 116 00:16:44,439 --> 00:16:50,370 convergent a simple example is now consider sequence 117 00:16:50,370 --> 00:16:53,450 (refer time 16:51) 118 00:16:53,450 --> 00:16:59,230 whether i start index with 0 or 1 it does not matter i am starting with 1 x2 is 1 x 119 00:16:59,230 --> 00:17:14,020 1 plus 1 by 2 factorial and so on so my nth element in this sequence is 1 by 1 plus 1 120 00:17:14,020 --> 00:17:22,010 by 2 factorial plus 1 by 3 factorial i think this is a well known series where does it 121 00:17:22,010 --> 00:17:32,080 converse to e but e is it a cauchy sequence it is known to be a cauchy sequence it is 122 00:17:32,080 --> 00:17:38,750 a convergence sequence is real line on real line where does it converse to 123 00:17:38,750 --> 00:17:55,480 so this sequence xn this converges to element e as n tends to infinity we know that this 124 00:17:55,480 --> 00:18:11,130 particular element tends to e but e is not a rational number so this element where it 125 00:18:11,130 --> 00:18:22,390 converges to is outside this space so you have funny situation you have a cauchy sequence 126 00:18:22,390 --> 00:18:26,320 if you apply the definition of cauchy sequence if you take any two elements as n and m goes 127 00:18:26,320 --> 00:18:31,280 to infinity you take difference it goes to zero that is very easy to show look at any 128 00:18:31,280 --> 00:18:32,940 book on real analysis 129 00:18:32,940 --> 00:18:37,900 you will see this proof it is just one or two pages of proof that this is a cauchy sequence 130 00:18:37,900 --> 00:18:47,330 but in this particular space it does not converge it does not converge and in this space i can 131 00:18:47,330 --> 00:18:55,180 find many such sequences i can find a sequence that is almost converging to pi but pi is 132 00:18:55,180 --> 00:19:06,730 irrational number pi is not there inside this space 133 00:19:06,730 --> 00:19:19,880 so likewise you know i have this sequence 134 00:19:19,880 --> 00:19:38,510 so this sequence that is 3 by 1 11 by 3 41 by 11 and so on it converges to not a rational 135 00:19:38,510 --> 00:19:44,610 number i can find infinite such examples where you have a convergence sequence you have a 136 00:19:44,610 --> 00:19:50,370 cauchy sequence but not converging to an element inside this particular space “professor 137 00:19:50,370 --> 00:19:55,580 - student conversation starts” those are rational numbers so this sequence is converging 138 00:19:55,580 --> 00:20:03,080 somewhere but it is not converging inside this space it will never converge inside the 139 00:20:03,080 --> 00:20:05,980 space 140 00:20:05,980 --> 00:20:11,210 so e does not belong to set of rational numbers that is what you are saying we know that in 141 00:20:11,210 --> 00:20:19,380 a real line this will converge to e see e is not the why these are all rational number 142 00:20:19,380 --> 00:20:29,590 sir we individually x is being as 1 by 1 factorial plus 1 by 2 factorial plus 1 by 3 factorial 143 00:20:29,590 --> 00:20:35,640 so you can always define one common denominator it is a rational number if it is just 1 by 144 00:20:35,640 --> 00:20:38,410 1 factorial it is a rational number 145 00:20:38,410 --> 00:20:43,770 no no all these are rational numbers i think we can talk about it little later this particular 146 00:20:43,770 --> 00:20:52,270 thing these are all rational numbers they are not irrational numbers so you mean to 147 00:20:52,270 --> 00:20:59,010 say that 1 by 3 may not be expressible but it is a summation of rational number rational 148 00:20:59,010 --> 00:21:04,200 number is whether you can write it as integer upon integer i can always write integer upon 149 00:21:04,200 --> 00:21:09,950 integer whether you can express it as a continued fraction 150 00:21:09,950 --> 00:21:14,230 we are not looking at that problem right now the true representation is integer upon integer 151 00:21:14,230 --> 00:21:20,940 i can have a common denominator for this it becomes a rational number you are confusing 152 00:21:20,940 --> 00:21:27,490 between its representations in this computer i am coming to that so do not confuse between 153 00:21:27,490 --> 00:21:33,720 the two so do not confuse one third with 0.33 do not confuse that with 0.33 “professor 154 00:21:33,720 --> 00:21:34,800 - student conversation ends” 155 00:21:34,800 --> 00:21:47,130 if this is true about q it is also true about qn i can define a product space which is qn 156 00:21:47,130 --> 00:21:57,210 n dimensional space my x can be qn i can take a space which is where do you get qn when 157 00:21:57,210 --> 00:22:07,581 i am doing computing in a computer i can deal only with finite dimensional vectors i can 158 00:22:07,581 --> 00:22:13,130 only leave with finite dimensional vectors and in computer you cannot represent many 159 00:22:13,130 --> 00:22:19,820 of these you know irrational numbers because computer has a finite precession 160 00:22:19,820 --> 00:22:25,760 if i take 64 beat precession the resulting number which you approximate as e actually 161 00:22:25,760 --> 00:22:31,900 will be a rational number something divided by i have to truncate right i cannot have 162 00:22:31,900 --> 00:22:38,280 a representation do you understand what i am saying in a computer whatever is the precession 163 00:22:38,280 --> 00:22:46,830 64 bit you know 128 bit you go to very high precession computer any number is actually 164 00:22:46,830 --> 00:22:55,570 represented as you know using binary 1 0 1 0 1 0 sequence and there is finite number 165 00:22:55,570 --> 00:22:59,400 of bits used to represent the number 166 00:22:59,400 --> 00:23:05,640 so that number will always be representable as a rational number something divided by 167 00:23:05,640 --> 00:23:13,210 something i truncate it so the point which i want to make is that incomplete spaces are 168 00:23:13,210 --> 00:23:20,380 not so you know when you work with computer you are working with incomplete spaces and 169 00:23:20,380 --> 00:23:24,980 we have to bother we have cauchy sequence which does not converge cauchy sequence this 170 00:23:24,980 --> 00:23:30,270 does not converge in a computer i will have a cauchy sequence which does not converge 171 00:23:30,270 --> 00:23:31,270 to a number 172 00:23:31,270 --> 00:23:36,470 no it is true value see for all practical purposes we say that well this is almost close 173 00:23:36,470 --> 00:23:45,870 to e but it is not e we take an approximation of pi may be you know correct up to 1000 decimals 174 00:23:45,870 --> 00:23:55,620 but it is not pi okay so we are working with this incomplete spaces and then let me give 175 00:23:55,620 --> 00:24:06,140 you one more example and i want to show a demonstration here of an incomplete space 176 00:24:06,140 --> 00:24:17,230 so my second example is set of continuous functions over minus infinity set of continuous 177 00:24:17,230 --> 00:24:28,290 functions over minus infinity to infinity this is my second example and i am going to 178 00:24:28,290 --> 00:24:34,110 construct a sequence in this particular vector space and what i want to demonstrate is that 179 00:24:34,110 --> 00:24:43,060 this sequence will converge to a discontinuous function i have a sequence of continuous functions 180 00:24:43,060 --> 00:24:51,700 converging to a discontinuous function so you are trying to solve some partial differential 181 00:24:51,700 --> 00:24:52,700 equation 182 00:24:52,700 --> 00:24:58,300 or some problem you construct the solution as a sequence of continuous functions or continuously 183 00:24:58,300 --> 00:25:03,640 differentiable functions the sequence might converge to a nondifferentiable noncontinuous 184 00:25:03,640 --> 00:25:16,980 function so you can have funny situations so my sequence here is this 1 by 2 plus my 185 00:25:16,980 --> 00:25:23,220 sequence here is a sequence of functions these are continuous functions defined over interval 186 00:25:23,220 --> 00:25:31,320 minus infinity to plus infinity this is a function sequence define so t goes from minus 187 00:25:31,320 --> 00:25:37,990 infinity to plus infinity my k changes k would be 1 2 3 4 5 i will get different functions 188 00:25:37,990 --> 00:25:56,110 for each value of k so i will get k goes from 1 2 and so on k goes from 189 00:25:56,110 --> 00:26:02,810 function sequence i just want to animate and show you what is happening so this is for 190 00:26:02,810 --> 00:26:12,370 k equal to 1 this is for k equal to 6 i am going to increment by 5 and see what is happening 191 00:26:12,370 --> 00:26:23,710 this is k equal to 11 16 and so on i just go on right i am going closer and closer towards 192 00:26:23,710 --> 00:26:35,030 this step kind of a function i am going closer and closer to the step function 193 00:26:35,030 --> 00:26:46,560 so if you do this i have gone only up to 100 if i 194 00:26:46,560 --> 00:26:53,660 do this by incrementing k much much longer much to a larger value this will converge 195 00:26:53,660 --> 00:26:56,110 to a step function 196 00:26:56,110 --> 00:27:03,560 so moral of the story is that i am starting with a set of continuous functions i am generating 197 00:27:03,560 --> 00:27:14,300 a sequence in this set but this sequence does not converge to element in the set the sequence 198 00:27:14,300 --> 00:27:26,220 does not converge to an element in the set so there is a problem so if what is nice about 199 00:27:26,220 --> 00:27:34,250 real line that every real line every sequence which is cauchy will converge to an element 200 00:27:34,250 --> 00:27:41,230 inside them ever cauchy sequence on the real line will converge to a number on the real 201 00:27:41,230 --> 00:27:47,420 line so in some sense real line is a complete set 202 00:27:47,420 --> 00:27:54,590 there is nothing outside it whereas set of all rational numbers is incomplete there is 203 00:27:54,590 --> 00:28:00,270 something outside and the sequences here seem to converge to something which is outside 204 00:28:00,270 --> 00:28:07,230 the space seem to converge or something which is outside the space so what is nice about 205 00:28:07,230 --> 00:28:16,200 real line its complete space what is nice about because real line is the complete space 206 00:28:16,200 --> 00:28:18,750 same thing is sure about r2 two dimensional vector space 207 00:28:18,750 --> 00:28:22,850 any sequence in two dimensional vector space will converge to the point in two dimensions 208 00:28:22,850 --> 00:28:36,390 any sequence in you n dimensional real rn will converge to element in rn but in qn there 209 00:28:36,390 --> 00:28:45,670 are holes you know so where the sequence be cauchy but it will not converge so this spaces 210 00:28:45,670 --> 00:28:52,420 you know in which all sequences converge within the space are called as complete vector spaces 211 00:28:52,420 --> 00:28:53,600 and these are special vector spaces 212 00:28:53,600 --> 00:29:02,080 so there is something different about the spaces in which so we move back to the black 213 00:29:02,080 --> 00:29:12,600 board so we want this nice property to hold even in the vector spaces so we call this 214 00:29:12,600 --> 00:29:17,530 vector spaces which have the special property as complete vector spaces or they are named 215 00:29:17,530 --> 00:29:24,460 after a famous mathematicians banach who actually founded this one of the founders of functional 216 00:29:24,460 --> 00:29:25,540 analysis 217 00:29:25,540 --> 00:29:39,870 so what is banach space so every cauchy sequence to converge to an element is space 218 00:29:39,870 --> 00:29:47,440 this word here every is important every cauchy sequence if i can find one sequence which 219 00:29:47,440 --> 00:29:55,110 does not converge the space is not a banach space every cauchy sequence should converge 220 00:29:55,110 --> 00:30:03,830 so the real line or rn or equivalently if you take complex numbers cn they have some 221 00:30:03,830 --> 00:30:06,380 very nice property they are all complete spaces 222 00:30:06,380 --> 00:30:11,900 function spaces need not be complete spaces set of continuous function we saw is not a 223 00:30:11,900 --> 00:30:16,590 complete space well in functional analysis you talk about completion of an incomplete 224 00:30:16,590 --> 00:30:25,230 space you add all the elements and then create a new space which is complete and so on but 225 00:30:25,230 --> 00:30:29,820 we do not want to go into those details right now i just wanted to sensitize you about the 226 00:30:29,820 --> 00:30:36,450 fact that even in a computer we are working with incomplete vector spaces and then you 227 00:30:36,450 --> 00:30:47,380 can get into funny situations in advance computing because of this incomplete behaviour well 228 00:30:47,380 --> 00:30:55,720 so far so good we talk about we started generalizing notions from three dimensions do not forget 229 00:30:55,720 --> 00:31:02,680 that we talked about a vector and then we said there are essential properties of a set 230 00:31:02,680 --> 00:31:09,500 which the two essential properties vector addition and scalar multiplications so these 231 00:31:09,500 --> 00:31:17,100 two things hold in a set then or if a set is closed under vector addition and scalar 232 00:31:17,100 --> 00:31:22,400 multiplication we call it a vector space any set so we freed ourselves from the notion 233 00:31:22,400 --> 00:31:24,080 of vector space which is just three dimensional 234 00:31:24,080 --> 00:31:29,320 we can now talk about set of continuous functions set of continuously differentiable functions 235 00:31:29,320 --> 00:31:35,190 set of twice differentiable three differentiable and you can so now how many such spaces are 236 00:31:35,190 --> 00:31:40,750 there infinite spaces then we said well we now that is not enough to have just generalizing 237 00:31:40,750 --> 00:31:48,430 of vector space we also need notion of length so we talk about norm right we talked about 238 00:31:48,430 --> 00:31:49,430 norm 239 00:31:49,430 --> 00:31:55,900 norm was in some saying generalization of notion of magnitude of a vector and we said 240 00:31:55,900 --> 00:32:01,880 there are so many ways of defining norms and a pair of a vector space and a norm defined 241 00:32:01,880 --> 00:32:10,120 on it will give you a normed vector space or norm linear space so this up to here fine 242 00:32:10,120 --> 00:32:17,800 now we need something more i need angle one of the primary thing that you use in three 243 00:32:17,800 --> 00:32:24,450 dimensions one of the most fundamental result in our school geometry or in three dimensional 244 00:32:24,450 --> 00:32:29,700 geometry pythagoras theorem and i need pythagoras theorem in all these spaces what i am going 245 00:32:29,700 --> 00:32:32,809 to do 246 00:32:32,809 --> 00:32:37,850 i need pythagoras theorem so i need orthogonality i need perpendicularity one of the most important 247 00:32:37,850 --> 00:32:45,290 concepts that you use in applied mathematics in modelling in physics in chemistry and every 248 00:32:45,290 --> 00:32:51,700 where orthogonality is very very quantum chemistry chemistry in the sense you might wonder where 249 00:32:51,700 --> 00:33:00,240 in chemistry so orthogonality is very very important and we need to generalize the notion 250 00:33:00,240 --> 00:33:03,930 of orthogonality and that is where we will start looking at in a product spaces 251 00:33:03,930 --> 00:33:16,650 we will start looking at inner product spaces so here the attempt is to generalize the concept 252 00:33:16,650 --> 00:33:21,670 of dot product “professor - student conversation starts” how do you define angle in three 253 00:33:21,670 --> 00:33:32,550 dimensions well if i am given any two vectors say x and y which belongs to r3 how do i find 254 00:33:32,550 --> 00:33:39,980 the angle between them so what i do is i find out excap which is a unit vector in this direction 255 00:33:39,980 --> 00:33:46,960 normally i take a two norm here well why two norm we will come to that why not one norm 256 00:33:46,960 --> 00:33:58,790 so this is something special about this two norm and why cap equal to and then we say 257 00:33:58,790 --> 00:34:17,679 that dot product that is x cap cos theta angle between these two vectors is just x cap transpose 258 00:34:17,679 --> 00:34:24,870 y cap this is the fundamental way by which we define angle between any two vectors in 259 00:34:24,870 --> 00:34:32,940 three dimensions now can i come up with something that we will generalize notion of angle in 260 00:34:32,940 --> 00:34:33,940 three dimensions 261 00:34:33,940 --> 00:34:42,970 when do you say two vectors are perpendicular in three dimensions dot product when dot product 262 00:34:42,970 --> 00:34:54,030 is 0 cos theta is 0 two vectors are perpendicular so i am going to peg on to these ideas well 263 00:34:54,030 --> 00:35:00,200 that dot product between unit vectors is used to define angle when dot product is 0 you 264 00:35:00,200 --> 00:35:08,780 call two vectors to be orthogonal and come up with a generalization in the product spaces 265 00:35:08,780 --> 00:35:16,869 of concepts of angle orthogonality and once the orthogonality you have pythagoras theorem 266 00:35:16,869 --> 00:35:23,310 i can talk about pythagoras theorem in any n dimensional infinite dimensional space of 267 00:35:23,310 --> 00:35:27,520 course it has to qualify certain properties what are those properties those are the properties 268 00:35:27,520 --> 00:35:33,300 of inner product space so now we have to start questioning what is characteristics of an 269 00:35:33,300 --> 00:35:41,820 inner product see we had three properties of magnitude what were the three properties 270 00:35:41,820 --> 00:35:42,820 of magnitude 271 00:35:42,820 --> 00:35:54,250 magnitude is always nonnegative for a nonzero vector and zero for a zero vector alpha times 272 00:35:54,250 --> 00:36:02,130 you get you know mod alpha gets multiplied to the norm and triangular equality likewise 273 00:36:02,130 --> 00:36:08,119 what are the essential properties of inner product in this which can be used to generalize 274 00:36:08,119 --> 00:36:14,460 in any other vector space those vector spaces are going to be called as inner product spaces 275 00:36:14,460 --> 00:36:20,710 because we are going to define a norm vector space in its additional structure is put called 276 00:36:20,710 --> 00:36:21,710 inner product 277 00:36:21,710 --> 00:36:24,190 “professor - student conversation ends” all these spaces which are describing till 278 00:36:24,190 --> 00:36:31,470 now we did not talk about inner product so now i am going to introduce something new 279 00:36:31,470 --> 00:36:35,540 which is the inner product space which will have definition of inner product what you 280 00:36:35,540 --> 00:36:41,370 release there are umpteen number of ways to defining the product and so the way of defining 281 00:36:41,370 --> 00:36:48,200 generalizing orthogonality is not unique and so we will see from our next lecture