Hilbert Spaces part 1

let's get started for today the topic today will be Hilbert spaces but before we reach that point I will remind you what we have done in the last two weeks and we also need a couple of steps before we introduce the hilbert spaces so what have you done so far so when I write last week actually it means the last two couple of weeks so we have introduced normed vector spaces this is what we did the first week then we moved on and considered panel spaces and actually we do not need the exact definition today so I will not write it on the board but the way we define pilot spaces was to require that all Kushi sequences were convergent that each Kushi sequence would converge to something in the space and you saw some examples of that so we solved the space of continuous function on bounders and closed in trial this is what we looked at already in the very first lecture and then last week we took another step because we introduced a space that we did not know about before namely you considered some P that was bigger than equal to 1 and then we introduced the space LP of N and we need that today so let me write that so this is the set of all sequences XK where XK are just complex numbers and then with the property that if you look at the sum of the absolute values of XK to the power P then we get something that is finite and turns out that this space is a Banach space we'll respect to and I'll just write that short will respect to and you need to tell what the norms should be so the norm is something we will write as XK and then with these parentheses around and then we put a little P below the norm symbol because we use that norm so many times and the way we define it is to take exactly this but in order to to get the norm axiom satisfied we need to take the peak root of that so we take the sum at K to the power P okay equal to 1 to infinity and then we take the P truth of that and actually you have some even simpler examples of panic spaces so let me not just write that as an example what I want to say now is the space that you know very well namely R in so let's look at some X that is in in and in order to make the notation a little bit easier than added last time let me put a bar below the X so it's clearer for you that now we are speaking about a vector so we will write it as X and then equal to x1 x2 all the way to xn and RN is actually also a Banach space and again you have to say with respect to which norm and actually we could you'll use all the norms that I have introduced here but I just want to state one of the norms namely the what we call the two norm the ball one way to take P equal to 2 so the norm of let me just write the X as X will a part below this will then be just the sum of X K to the power 2 K equals 1 to N and then we take the square root of that so this is what you know already from mathematics one or whatever cause you had at the beginning of your education the reason that I want to state this game is that if you think back about what you did at mathematics one or whatever you had at that time you also know that are in has another concept associated to it namely there's a inner product a scalar product so this is something I'll just remind you so RN has an additional structure namely the structure of an inner product and some of you might not have called it inner product some of you might have used the word scalar product instead instead and the way you define this is that you take the vector X with the bar below and then you put a dot and then you write another vector Y and the outcome of this is something that you have written as the sum from k equal to 1 to n of xk x YK so you simply take the coordinates multiply them together and then make the sum or all coordinates so this is something you know very well the reason that I want to take this again is that you can see this is closely related to the norm so what you see here is that if you take the inner product of a vector X with itself when you get the sum of X K to the power 2 which is exactly what we have inside here so in other words you get the norm of X to the power 2 so this just says that's the inner product between a vector and itself this is equal to the length to the power 2 of that vector now this in a product that you have on in this is actually one of the main ingredients in the theorem for RN it's one of the main ingredients in all linear algebra it allows you to speak about or analogy of vectors so as soon as you have an inner product you can speak about what does it mean that vectors are perpendicular to each other so what we would like to do now is to take the step of generalizing the inner product from something that you know about on are in and now we want to do exactly the same on any vector space so let's do that so we'll make a generalization to arbitrary vector spaces so let me define the concept for you so the starting point is to choose any vector space and I would like to put an extra word here I would like to say that it should be a complex vector space so a complex vector space this means that the scalars that you are multiplying on your vectors in the space they might be complex this is not the case for the space are in so our n is not a complex vector space is a real vector space so what I'll do is to write down the exact definition for complex vector spaces and then after that I'll see if I can find some red chalk and then I'll write here real and then I state these few changes that are necessary in case we need to look at a real vector space most of the spaces we will speak about in this course are actually complex vector spaces so all the new spaces I will introduce will be complex vector spaces the only real vector space you'll meet is actually the end that we have down here so a complex vector space then we want to define an inner product and you'll do it a little bit in a similar way to what we did as we introduced a norm so we will say an inner product this is a mapping so you remember as I define the norm I was saying a norm is a mapping that to each element in the vector space associated a real number so we will do the same here we need to find a name for this mapping and the way it is to note it is by a little bracket that turns this way and a little bracket that turns this way and then we put two dots in the middle and then a comma so this is not a smiley face or anything like that this just means that these are the places where you put in the variable so you have two variables to put in exactly like you have two variables to put in here you have to put in an X and you have to put in a Y so what we are putting in here are elements this should be something that belongs to V this should be something that belongs to me so it's a mapping the curves from V cross V in the sense that we have two errors in V and then two each such pair we associate a complex number and here you see the difference the scalar product we have over there does not give us a complex number it gives us a real number so this is the difference between the complex and the real vector spaces but I write that down in details later so this is a mapping but we want this mapping to act in the same way as the inner product we have here that means we want this abstract mapping to have some of the same properties as we know from the inner product so we will say some properties the first priority is that we actually have linearity with respect to what we put in in the first bear record so that means if we put in some alpha v plus beta W and then you take an inner product with some elements you so that means alpha v plus beta W displays the role of what we're putting in here and the you place the role of what we're putting in here then we want this to be the same as just to take the Alpha and then the inner product of the V with u plus and then we take the beta and then we take the inner product between W and you so what this actually says is we want the inner product to be linear it should be a linear function of what goes on in the first variable so we want this to happen for all choices of these vectors V W and you so for all of them belonging to V and you also want it to happen for alpha and beta belonging to the complex numbers the second one is a little bit easier it says that if you look at the inner product between V and W then you know if you look again at this situation whether you take in a product of X with Y or you take Y with X this will give you exactly the same number so you can exchange the order of x and y if you want because here you just exchange two numbers that you're multiplying so this does not make any difference here in the setting of complex numbers this is different so what we require is that this is the same as when we take it in the opposite order but then you take a complex conjugate and then so this is a change compared to what you know already this is actually five to the third one is not something you have really thought about in the setting of RN but we will see the role of it very soon so what it says is that the inner product of V with itself this is always a positive number this is of course a property that you know from this situation because because the inner product is equal to X star X K times XK and then the sum which is equal to express we have here so we know that we get a number that is greater than or equal to zero but we do not know why this is so important but here we put this as one of the conditions and we also want maybe as would say we want this for all V in the space and let the inner product it's equal to zero this is exactly the same as e being zero so these are the conditions that we will put on an inner product and I have more or less already given you the first example of that so RN would be the first example so I mean we do exactly what we're used to and then we look at this definition and we check that all these properties are satisfied I would not do that what I'll do instead is to look at the complex case because this is what is new for you so CN is also a vector space with an inner product and then I need to tell you what is this inner product and the way we define it is X Y and we take almost the same as what we have here except let me put a bar under YK so let's check that this actually satisfies all the conditions that we have here so let's call it a proof but we only look at it for the case of CN so you can see the situation is very similar to what we had as we looked at normal vector spaces normed vector spaces we also had a condition one two and three and we actually had something else that we should check before we started the proof namely you see here we need to check that we actually have a batting from V cross V into the complex numbers this is the first to locate so this is said it's right here if we if you call this condition for somehow the condition zero because this is not this is not staying among the one two and three then we just look at what we have here we'll get the definition of X in a product with y and then you see all what we have here is a product and a sum of some complex numbers so what comes out here will be a complex numbers so this is something that belongs to C so this is the first part to check so now you have to go through 1 2 3 so let's take them one by one so let's look at at the linearity so this is alpha v plus beta W in a product with u now where we're dealing with sequences I think the natural notation would be to say alpha times the vector X plus beta times the weight vector Y inner product with some vector C so let's look at this so the definition says the definition that is stated here says that this means we are looking at a sum starting from k equal to 1 to n and then you have to take the cake component of the first vector and the cake component of this is just the elephant times the XK plus the beta times the Y K so we have to take this and multiply with the cake component here where you remember that everything in the second coordinate should be complex conjugated that means we are looking at GK complex conjugated and energy ok you look at this and you say this is just a finite sum of some numbers so we can manipulate with this exactly like we used to so you see here you can actually split it in something that has to loot with Alpha X K seek a complex conjugate and something that has to do with this and that so the first term will be this times CK complex conjugated here the alpha does not depend on K so this is something we can take out so this is the same as alpha and then the sum k equal to 1 to n xk CK complex conjugate and for the second term we get to the beta times the sum a equal to 1 to N and then we get YK seek a complex conjugate and if you look at this this is exactly the elf at times what we have here by the definition this is simply the inner product between X and C and then we get plus beta and what we have here is exactly the inner product between Y and C so that means this verifies the first condition the second condition means that we look at the inner product between two elements let's just now call them inner product between x and y so again if you look at the definition we just take it from here so this is the sum k equal to 1 to n XK YK complex conjugate and we want to manipulate with this such that we get something with where the Y is in the first coordinate and the X is in the second coordinate and the way to do that is simply to put a big bar or everything and then look at what is inside here so if you have a big bar or everything we don't need to conjugate the YK and I can even change the order you want because these are just complex numbers so this is YK and then I want something that has to do with XK so if I just write XK it is not correct because the XK will then get conjugated here which is not what we have here but if you take the complex conjugate it inside this then we take the conjugate of XK and we conjugated one more time and this will give you the XK back so this is exactly the same as what we have here and if you look at what is inside here this is exactly the inner product between the Y and the X so this is what is inside here and then you have a complex conjugate and everything so you see now you can change the order and the price to pay is that you have to put a complex conjugation and everything so this is okay as well now you might actually have wonder why did I choose to put this complex conjugation here but you can see that now because when we look at the third condition we need to look at the inner product of an element with itself so let's look at X inner product with X and by this definition this is then what you see you get here is X K times X K complex conjugate which is exactly the absolute value of XK to the power 2 so you only get that this is positive because the definition of the inner product actually has this complex conjugation on the second interest so this shows that you have the first property and then we need to show what does it mean that it is equal to 0 and then we just look at it and say it is given by this expression and this is the sum or non-negative numbers so this being equal to 0 is exactly the same as to say that XK is equal to 0 for all K and XK be equal to 0 for all K this is the same as to say that X is simply the 0 vector so this verifies the the third condition that we had here so that means we have indeed that CN is a complex vector space and that this defines an inner product on the space let's do a small change compared to what we are doing in the definition so if you're looking at the definition here you see the linear combination is in the first coordinate what happen if you put the linear combination in the second coordinate instead so in other words what happened what happened if you're looking at an inner product between and now we go back to just the setting of abstract vector spaces we look at the inner product between V and some alpha W plus beta u so can any of you tell me how can we get an expression for that Alexandra you swap them yes because then you have the linear combination in the first entrance and then you can use the first one so let's do that but let's do it quite fast so you say we swap them so we get there alpha W plus beta u and then the V and then you the price to pay is that you have a complex conjugation on this now the idea is that when the linear combination is in the first entrance we can use the first rule to split it by linearity so this is the same as alpha and then you have W will be plus beta and then we have you will be and then we have a complex conjugation on everything here so what comes out is that the complex conjugation and everything will go down to be a complex conjugation on the Alpha and then we have a complex conjugation on top of U V but then we can just remove that complex conjugate again by swapping the order again so then we get V and we get W and for the second term exactly the same happen so we get the Pater complex conjugate and then we get you we complex conjugated and then again we swap the order and we remove the complex conjugation so this is then V with you so you see this is dangerous this is really a point where you have to be careful because intuitively if we didn't look carefully at the definition you'll probably say that it does not matter whether we had this linear combination in the first entrance or in the second entrance but what you see here is that if you have the linear combination in the second entrance you can still split it out using some kind of linearity but you have to remember to complex conjugate all the coefficients that are coming out so remember that okay let's again look at something that you know so the way we will develop the theory is actually to play pin point with what you know and what is new so we always go back and look at something you know and then we move it to the new setting so we have been speaking about RN but that's being about R 2 so in R 2 what does it mean to have an inner product in R 2 this means that you have a vector X that is sitting here and you have a vector Y that is sitting here and you have an angle between these two vectors then you know that the inner product X Y this is actually the same as the length of the vector X times the length of Y times cosine to the angle theta so this is probably something you saw already in high school so what this means is that if you're looking at the absolute value of the inner product then this is equal to the length of X and the length of Y times the absolute value of the cosine an absolute value of the cosine is at most equal to one so in other word this absolute value is smaller than or equal to just the length of X sine length of Y which if you write this in terms of the inner product then this is the same as just the inner product of X with X to the power 1/2 so you see this is what you get from what we already wrote down here so you can relate the inner product at the length of the vectors and in the similar way this is just the inner product of Y with Y to the power 1/2 so this inequality that the absolute value of an inner product is smaller than or equal to this is something that we can also state in general vector spaces and actually if you take the proof of this and you look at it in the setting of RN then exactly the same proof can actually be used in arbitrary vector spaces because if you go and look at the proof then you will see that the rules that have been used again and again in the proof are exactly the ones that are stated here so I will not go through this proof because this is something that you can actually go back to your notes from mathematics one and then you can just see how to how to prove these steps and you can see that it's exactly the same we have to do here so the similar result in vector spaces this is formulated in the book and say theorem and this is something that you will use repeatedly in this course it actually has the name Cruces Watts inequality so what it says is that if you're looking at an inner product between V and W and you look at an absolute value then this is smaller than or equal to the inner product between V and itself to the power two times the inner product between W and W to the power 1/2 so I will not prove this because the proof is the same but I will show you how we can apply this to connect the norm and the inner product so it actually turns out that no matter which vector space you're looking at as soon as you have a complex inner product then you can actually define a norm on that space and that means that the setting of spaces within a product is actually a sub setting of what we looked at in the very first lecture namely the vector spaces with a norm so this is a very important observation so I'll give you the full proof for that so let's formulate the result first so what it says is that if you have a vector space with an inner product so we take the inner product and we denote it by this triangle notation then you actually automatically have a norm on the space so we can define the norm and the way we do it is exactly like we did here so the norm should be defined by taking the inner product of an element with itself and then take the square root of that so let's see how we can prove this so I think it's most convenient that you have the definition of a norm can all get so what we have to do is to again you see we have three points to discuss but we have the extra point that up here in the proof for CN you see I introduced this point zero and the fence error is also something we have in the in the definition of the norm because this is that a norm should be a mapping from the vector space into the real number so this is something we always need to remember to check so we check these three conditions but we also check that the norm actually Maps the space into the real numbers so again if you call this point zero then we need to look at this and we need to check that this belongs to our and the reason that this belongs to our this is because this is the way you have to find it you see it is a positive number or maybe zero so this is a consequence of three so this is okay so let's go through the points 1 2 & 3 here so let's look at the first one we need to show that the the norm of V is greater than or equal to zero but the norm of V this is just the inner product between V and V to the power 1/2 and again if you look at this then this is positive so this is ok and what does it mean that the norm is equal to zero this means that this is equal to zero and by the condition that we have here this is the same as to say that V is equal to 0 so both these conclusions are by using the definition for 1 1 and then simply the first part then let's look at the second one else we need to look at this in norm and then we need to show according to what we have on the slide we need to show that this is the same as the absolute value of alpha times the norm of the V and the only way we can do that is to use the definition of the norm and the division of norm says that then we have to put in the vectors and take the square root in the inner product so this is alpha v alpha V and then to the power 1/2 so we want some how to get the ELSA out and for this alpha there is no problem we can just take it out because this is what the first condition tells us that this is allowed Elsa in the first coordinate can always be taken out so this Elsa goes out without any problem what about the second alpha this is sitting in the second entrance of the inner product so the calculate meter stick shows us that whatever we have here can also go out but we need to put a complex conjugate me on that so this comes out as alpha with a complex conjugation and then what is left is the inner product will be and itself and then take the square root of this as I times alpha complex conjugate this is the same as alpha to absolute value to the power two and what we have here the inner product will be and itself this is according to this this would be the same as we to the power two so we need to take the square root of this and then you can see to take the square root of all this what comes out is the alpha times the length of the V so this follows from from our definition of the inner product what about the third one so in the third one this is the triangle inequality so we need to look at the norm of V plus W and you need to show that this is smaller than or equal to the norm of V plus the norm of W it turns out that instead of working with this as a state here it is better to work with to the power two because then we avoid to put the square root all the time you see here I had to run around with a square root all the time so in order to avoid that let's look at at this to the power two and make the calculation for that and enough that we can always take the square root so the norm of this to the power two again according to the definition this would be the inner product of the vector with itself so this is the inner product of V plus W V plus W and now you see you have a linear combination in the first entrance so you can use again the first property we have here to split it out as two inner products so you can say this is the inner product of V will plus W plus the inner product with W and then V plus W and now you are left with something where you have a linear combination in the second entrance but the coefficient is 1 times V plus 1 times W these numbers are real they are not complex so this means that you can split further here so we can say this is the same as just the linear combination I'll actually write it here this is the V in a product with v this one with this one and then the inner product will V and W and then you split this in the same way so this is W in a product with V plus W in a product with W now what we are calculating here is something that is greater than or equal to 0 because this is what we observed already here that the norm of any vector will always be bigger than or equal to 0 so all what I have written down here is greater than equal to 0 so this means that if I want I can put an absolute value here it does not change anything but it is a very good idea to put this absolute value because we have something called the triangle inequality that says that if you take the absolute value on a sum of a lot of terms this is smaller than or equal to what we get by taking the absolute value of the first class up the value of the surgeon plus and so on so all we have here is smaller than I you consume absolute value V when we plus V with W plus W with V plus W will W and this is something we can estimate the first one is the inner product of V with itself again by the definition of the norm this is the same as norm of e to the power two this inner product we don't know what it is but Christi Schwarz tells us that this is smaller than equal to V the norm of V times the norm of W because you see what we have here you get the inner product of V with itself to the power two this is exactly the definition of the norm of V and this is the exact the definition of the norm of W so this is what we get for this term if you look at this we get exactly the same we just get it in the opposite order but we still get just the norm of V times the norm of W so actually at the end of the day you get this but we get it with a factor of two and then we have this term and this is norm W to the power of two and then we do something that you did in high school but of course you don't recognize it here because of all these strange symbols but what you have here is exactly the same as the norm of e plus norm of W all of it to the power 2 because this to the power two means this to the power 2 plus this to the power 2 this would give us these terms and then 220 stands bet and this is exactly what we have in the middle so that means all these manipulations have given us that the norm of this to the power 2 is smaller than equal to this to the power 2 and then we just take the square root of M everything so this gives us that the norm of V plus W is smaller than equal to the norm of V plus the norm of W and this is exactly the third condition that we wanted to check we wanted to check the the triangle inequality that we have here so using all these manipulations we have actually done that so I think by now you must be convinced that this is a good definition of of an inner product because the definition we have here play very well together with the definition of the norm that we already have now I promised to do something for you and before I forget if let me see if I can find some red chalk so we can note how the definition of the inner product need to be changed if you look at real spaces so maybe the blue is the best one so what happened if this turns out to be a real vector space so let's just mark it here no this is not good I can see that on the board so let's take the orange instead so what happen if you are dealing with a real vector space instead then an inner product will still be a mapping that takes a couple of vectors and maps it to some numbers but in that case we want the outcome to be a real number instead of a complex number and this also means that when we are looking at linear combinations when you have a real vector space this means that the coefficients alpha and beta are real numbers instead of complex numbers so here we need to change and say we put in real numbers instead but besides this this is just linearity so this different the definition as such is the same now here since this is a real number whether we make the complex conjugation or not does not change anything so in the real case this will just be equal to the inner product W V so we simply remove the bar in the real case the last part of the definition is exactly the same so you see there are these minor changes and you just need to pay it whether you're dealing with a real vector space or a complex vector space and as I say all the vector spaces we are dealing with in this course are actually complex vector spaces so this is just made in order to really to make our NP a special case of what we're doing here now before we take the break I would like to show you a very important statement about the inner product spaces so the key results in this section it's actually the one that has the name theorem 4.1 for that comes right after what we have discussed so far so the question is how much is new here and how much do we already know so you see it has four points and if you look at the first one this is exactly the Kosovars inequality so this is exactly the one that we already stated and I think I took it away already we needed the space but there's no change this is exactly because it's worth that we already looked at so the new path is from two down 2.5 so all these on you and all of them will be something that you will prove during the exercises not all of them today but during today and doing some homework and during next week I think we'll get through all the proofs of these statements let's look at the first one let's look at the free pint for first let's do it here so we write down some comments to hear em 4.14 so let's look at the point three and four so you can see what they do is to relate the norm and the inner product so what they show for example if you look at four this is in the case we have a real vector space what it says is if you can calculate all the norms that are stated here then we can actually calculate the inner product of V and W so that means if you know the norms of all vectors then you can actually calculate back and you can find the inner product if you know get free this is exactly the same if you can calculate all the norms that are stated here then you also know the inner product between V and W so what this says is that you can actually recover the inner product if you know all the norms so you think about this this is exactly the opposite of what we did so far so what we did so far was to say let's take the inner product and then we define our norm can you see this is what happened here in the very first line we start with an inner product and then we define the norm by the inner product so if you have the inner product you know how the norm is by definition and the statements that we have here point three and point four needs tells us that you can also go back if you know the norm then you can go back and tell what the inner product is there's one more of them that I would like to say just a few words about and that's point to the parallelogram rule so it turns out that there are many vector spaces where you don't have an inner product we have just defined vector spaces with inner product but there are a lot of ways of norm vector spaces where you don't have an inner product so what this tells us is that the definition we have here this is a very nice definition when it is satisfied but there many cases where these conditions do not work we cannot make an inner product that will help us but what the theorem tells us is that if we have a case where there is a norm coming from an inner product then we actually need to have the parallelogram law satisfied so that means if you are in a case where you have a norm on the space and you do some calculations and it turns out that the parallelogram law is not satisfied then you know that the norm does not come from an inner product so the role of this is that if the parallelogram rule if this is not satisfied then actually the norm we're dealing with does not come from an inner product and this comment is something that would be helpful helpful for you when you make your homework because it turns out you have two exercises that are called F 4 5 & 6 2 and in these exercises you actually use exactly this observation so please remember that I don't know if you have any questions or if not yes you have a question this one you see the assumptions in the theorem is that you look at a vector space that has an inner product and then you have associated norm so it is about a norm that comes from an inner product so let's take out right now and let's say that we're back in the usual time about 12 minutes something like that you

Channel: DTUdk

Views: 93,092

Rating: 4.9113574 out of 5

Keywords: LearningLab, DTU, Lecture, with, Ole, Christensen

Id: jWkzBaJDSmY

Length: 50min 18sec (3018 seconds)

Published: Wed Feb 20 2013