 Welcome to module 59, topological vector spaces. Throughout this section, we shall assume that k denotes either the real field or the complex numbers field C. By topological vector space, we mean a vector space V over k together with a topology on the underlying set V such that the algebraic operations of addition and scalar multiplication etc. are containers. So, this is not yet a definition. I will give you precise definition in a moment. Okay, just similar to a topological group. More precisely, a topological vector space consists of an order 5 tuple, quintuple vector space V, a set V, a plus operation, a dot operation and a special element 0 and a topology tau. The first 3 V plus and 0 constitute an abelian group wherein 0 is the 0 element of the abelian group. The dot denotes a scalar multiplication k cross V to V. It is not from V cross V to V. It is not a binary operation. It is a scalar multiplication and tau is a topology on V. So, they have some interrelation here. What are that? First is a prior algebraic. The scalar multiple alpha beta of V is associative. That is the first condition. And if you multiply by 1, it is identity. 1 times V is V for every V. So, this is the scalar multiplication property of vector space, you may say. Alpha of U plus V is alpha U plus alpha V is distributed V of the scalar multiplication. So, 1, 2, 3 actually constitute what is the meaning of vector space. I am really defining it here. That is all. The map V cross V to V given by U V equal to U minus V is continuous. This is similar to what we have done in the topological group k. U V or x y equal to x y inverse is continuous. So, instead of multiplicative notation, we are using now additive notation because we started with this one is an Abelian group. So, additive notation is preferred here. So, it is U minus V is not U plus V. U plus V as well as V equal to minus V both are continuous is equivalent to one single condition. U V equal to U minus V is continuous. The map, the scalar multiplication k cross V to V given by alpha is going to this alpha dot V. This is the dot that is continuous. What is topology? V as a topology, k is the standard topology, equivalent topology, k cross V is a product topology. Under that, this map must be continuous. So, k is taken the equivalent topology and k cross V and V cross V etcetera here are a given the product topology. So, for those who know what is a vector space already here which we have used several times, the multiplication, scalar multiplication and addition they are continuous is the easy way to remember that it is a topological vector space. Vector space in which there is a topology with respect to which algebraic operations are continuous. This is the way to remember instead of all this list here. Notice that if you just ignore the scalar multiplication, what we have is a topological group which is abelian. So, in particular all the results that we have proved about tau will hold in this case also except that we have to carefully replace the multiplicative notations with additive notation everywhere ok. Now, what we expect is something more should happen stronger results should happen because, because existence of this continuous scalar multiplication. So, that is what we anticipate and well soon we will see that that is what is true anyway. So, let us now recapitulate, reorient our notation according to these vector space notations here alright. So, a and b are any two subsets of the notation ab we used will now be written additively that ab for topology group was alright now it does not make sense for us. So, you should write a plus b which just constitutes all points little a plus b where little a and little b are respective in a capital A and capital B. Similarly, a minus b which corresponds in the earlier one it corresponds to ab inverse right. So, at the inverse is now here minus a minus b where a is inside a and b is inside b. So, minus a which just a inverse is all that minus a where a is inside a ok. So, these are the proto type of whatever we used earlier in the case of just topological group and wrote in the multiplicity notation there. But now we have another multiplicity notation here only scalar multiplication namely a must be a subset of k and b must be a subset of v then we have ab this makes sense all points alpha dot v where alpha is inside a and v is inside b. So, as far as possible we will use these Greek symbols for the scalars or t, s etcetera. But ab, cd or v etcetera we will be denoting the vectors. We shall dispense away writing the whole five to pull all the time that is our standard practice of shortening the symbols ok notation. We nearly use the phrase v is a topological vector sphere with the understanding of addition scalar multiplication at the power g and the zero element etcetera. We shall also dispense away with the symbol dot. Now even dot we will not write just simply write its alpha v. Normally you know when you have a vector space you write the scalars on the left the vector on the right even that is liberally you can use v times alpha also if there is no confusion ok that is also allowed. But its standard notation is to write the scalars on the left. Now you start doing some topology let o be a neighborhood of zero in v. Then for any sequence of positive real numbers r n converging to infinity we have the entire vector space v is covered it is contained inside the union of r n times o where n rings o 1 to infinity such a thing is obvious in the case of r or r 2 or r 3 and so on ok. So why this is obvious for a arbitrary topological vector space I will give you a minute think about it. So you have an answer some of you must have have an answer. So congratulations if you have done it correctly here is the answer fix a v belonging to v I must show that it is inside r n of o for some o what is o? o is fixed o is a neighborhood of zero inside we get r ok. Look at the map from r to v given by r going to r times v this v is fixed ok r goes to r times v this is a continuous function why because scalar multiplication is continuous right. Therefore inverse image of an open set see v o is an open subset of v right. The inverse image of open subset and ref this will be an open subset of r what happens if r put r equal to 0 0 goes to 0 and 0 is inside o therefore u is a neighborhood of 0 in r then we know that if r n is in n is large since r n goes to infinity 1 by r n will be inside u right because u is a neighborhood of now 0 what is the meaning of 1 by r n is inside u f of that is what f of that will be inside o f of that is nothing but v by r n to v by r n is inside o means v is inside r n times o ok that is all you see. So the scalar multiplication is a devil here it is going to give you a lot of things which you have missed in a metric space it would not be as strong as having a metric space ok but in some sense it is stronger also since some sense it is weaker also so that is the game we are going to play just that is why I stopped to you know to give you a little more time to think about what is happening. The scalar multiplication is going to play a lot of you know role here as compared to just the attitude which is there attitude we have seen something about topology of a group but this is the extra thing right ok so such a thing you could say in a in a topological group now I will introduce a number of definitions which will involve the scalar multiplication ok a subset b of v is called convex subset you see this was always possible in a vector space I am not doing anything else at the same definition no change of definition here ok if u and v belong to b should imply the line segment 1 minus t times u plus t times v 0 less than equal to t less than equal to 1 the entire line segment is inside b that is the definition of convex subset ok but in the notation that we have introduced it is same thing as saying that 1 minus t times b plus t times b is a subset of b so I have put elaborate notation here because you may confuse it for something else for me there is no confusion the singleton t times b is same thing as just tb this we have been using right so I have used that one also tb is nothing but singleton t times b ok so that is all the shorter notation here these brackets are not written here but this is by definition singleton 1 minus t times b plus singleton t times b so that is contained inside b same thing this is you know for each point here and each point here ok for each point here and each point here u comma v so that line segment is inside b same thing as this one once you have convexity you can talk about local convexity we say v is locally convex if there is a local base at 0 consisting of convex neighborhoods take any neighborhood of 0 inside that you must have a convex open subset around 0 contained inside the given number that is the meaning of there is a base the local base consisting of locally convex open subsets convex open subsets that will be called then we are called v is locally convex actually you should define it for every point but we do not need to bother about other points once something happens at 0 you can translate the whole thing any topological property which holds at one point will be true for all the points so that is the property of topological groups in general so it is true for vector space also okay note that this will automatically imply the same thing same thing means what it is convexity with all the points of local convexity so one point is enough for the definition like Rn that is locally convex there are many topological groups which are not locally convex so that is why we want to define this one a subset A of v is called balanced okay this is something new you might not have defined we might not have met with this one standard study of vector spaces and so on okay it is called balanced if alpha b is contained inside b for all modulus alpha less than 1 this mod alpha less than 1 becomes crucial okay when you take complex numbers that will have different meaning you see if real numbers it's just a line segment right complex numbers it's a whole disk so geometrically they will be they will have different meanings okay so balancer means that let me illustrate this one it will go over because this is a new concept okay you should not confuse it with something which you may think is the right one suppose b is a balancer set okay if b is non-empty then 0 must be inside b do you see why because look at this one look at the condition condition says that mod alpha less than 1 alpha b times alpha b must be inside b if you put alpha equal to 0 that's valid right so 0 must be inside b okay only thing is I need to have some vector here if b is empty this is automatically satisfied empty set is balanced to find but then you can't say 0 is there right so only non-empty I have this non-empty then 0 is b more generally take any vector v is b and b is balanced right okay okay the entire line segment minus v comma v will be inside b okay see once v is there I can take alpha equal to minus 1 so minus of v will be also there and minus v and plus v is there you know I can take any other number also here between t and minus t between 1 and minus 1 so all those multiples also there that means the entire line segment is there okay one element is there then the line segment minus v plus v is there so this is so far I have used only real numbers next if suppose v is a vector space over complex numbers right suppose if case your complex number and v belong to be implies the same thing now I can take the entire disk d2 multiplied by v that's also a disk because multiplying by v is a isomorphism is a homomorphism right unless v is 0 okay non-zero vector you have to take if you take non-zero vector it will be 0 that's also fine no problem okay so entire disk d2 v is contained inside b okay so this balanced the balancedness the definition of this word balanced is an indirect way of bringing the open walls inside a matrix space you will see that balanced subsets balanced neighborhoods and so on they are going to play the role of the open walls in a head metric space okay that is the whole idea of this one so I have used to be for balance but they are also you may say they are prototypes of balls in metric space okay now let's go ahead with some more definition now I am going to introduce another term here though there are no metric spaces we are going to define some notion of boundedness so it's again another backdoor way of bringing boundedness though we don't have metric okay so pay attention to this definition a subset b contained inside v is said to be bounded if for every neighborhood u of 0 there exists m positive such that this s is bigger than m implies b is contained inside s u s u s is a real number s u is like a expansion or contraction of u okay s times that open side that's it the whole thing is I started with u as a neighborhood of 0 so 0 will be always there all the vectors will be stretched by s times that right that's why if s is bigger than 1 then it will be larger smaller than it will be smaller but whatever it is s is bigger than m is important here this is not a typo this is not mod s this is s okay real numbers only here s bigger than m should imply b is contained inside s u there must be some m how small how big I don't care so that is the meaning of boundedness which is a very strange definition you will see slowly you will realize that this boundedness is actually stronger than the metric boundedness for an ordinary space okay equivalently let us look at the other way around here for every neighborhood u of 0 there exists an epsilon positive such that delta is less than epsilon should imply delta v is contained inside u so starting with a b by contracting b smaller and smaller you can bring it inside u so that is the meaning of this boundedness why this is equivalent to this one you have to just invert it you see b is contained inside s is same thing as 1 by s is contained inside 1 by s of b is contained inside u so delta equal to 1 by s so here also you can invert all of them and instead of s bigger than m it will become you know 1 by s 1 by m is less than 1 by sorry 1 by m is less than 1 by s right so this 1 by s is your epsilon 1 by m is your delta that is that is the way you can interchange these two so this is the meaning of something is bounded okay so let me give you an example here consider v equal to the complex numbers as a one-dimensional complex vector space don't go for you know very large here it's you you have to understand it only balanced sets subsets of v are empty set open or closed balls with center 0 and the entire space v so these are they all balanced now balanced that they are open balls only thing is they are centered at 0 by translating them you will get all other balls you see so that is why this definition however consider v equal to r2 now as a vector space over r itself not as complex vector space okay as a two-dimensional vector space then there are many more balances subsets such as any line segment with center 0 okay any line segment with center 0 as it's center means which you have to minus r to plus r your rotate okay it need not be r it could be line line it's lines passing through the origin that's all okay or union of such segments for example you can take x axis and y axis that will be balanced or x y equal to 0 x plus y equal to 0 and x minus y equal to 0 as well as x axis y axis and so on star-shaped thing wherein all the subsets check they must be equally distanced from the from center okay so that's why balanced the word balanced it is just that so attention should be paid specifically whether in the given context we are using k equal to r or k equal to c okay geometrically they have some different meanings here so in the k equal to r it is not exactly balls instead r it is centring intervals but but in j r2 r3 and so on this something different okay whereas convexity it is same thing no problem convexity does not depend upon whether you are using complex numbers or real numbers because the definition of convexity is only real so t times v plus one time one minus t times u where t is a real number between 0 and 1 you don't you don't use complex numbers at all so convexity is the same thing for whether you are treating a complex vector space or real vector space so I come back to an example of this boundedness also okay the world of caution about the boundedness concept that we have introduced here for topological vector spaces it is not dependent on any metric or norm there is no metric or norm we have used everything is inside a topological vector space concept also it is somewhat stronger than the usual boundedness concept in a metric space so how do you state that maybe you start with a norm in your space that is a topological vector space right any norm in your space is a topological vector space if the norm you get a metric right but the metric you change it by using this rule d of x y equal to here I have used only complex numbers or real numbers but we can do it for any norm in your space of norm of x minus y divided by 1 plus norm x y d x y divided by 1 plus d x y that is the meaning of this right then this is always a bounded metric gives you same topology so the original topological vector space will be a topological vector space with this topology or same topology right only metric has changed it is not a linear metric it is not norm okay but if you use this metric the whole of r whole of r n all of them will be bounded because it is bounded by 1 whereas immediately just look at r and the boundedness concept that we have introduced okay it is not bounded in that sense neither in the standard sense it is bounded r is not bounded when r will be bounded take any neighborhood then you must be able to collapse the whole of r by one single scalar inside that neighborhood that is not possible because no matter how much what scalar you multiply no matter how small epsilon times r is the whole of r itself right so r or any other vector space are not bounded in that sense okay so I promise that in the second part we shall consider another concept of boundedness for metric spaces which is even stronger than this this metric this boundedness concept okay so that will be called totally boundedness which is which is the name is quite justifiable so that will be done in only part okay so let me have one more simple lemma before giving a break here for v belonging to v and alpha not equal to 0 we have the linear homeomorphism respectively called a translation operator tv and a multiplication operator m alpha for alpha m alpha for v tv so these are two two homeomorphisms okay of course if v is 0 the tv will be identity but for this one you have to assume that alpha is not 0 okay otherwise multiplication will collapse the whole thing it will not be a homeomorphism so how it is defined tv of v is equal to u plus v m alpha of v is alpha times in particular for any subset open subset u of v and any subset x and y of v and k and so on you will have this similar theorem that we had earlier namely x plus u equal to u plus x here because this is commutative why you I do not want to write you why because scalars should be written on the left only they are open some sets of t okay so this is this is something which we have done similar to just the case of topological groups what do you think this is new because then alpha is not 0 there is a multiple inverse also m alpha inverse will be inverse of m alpha so m alpha is also a homeomorphism okay this is the extra thing that is happening here okay so lemma is obvious why because you have to just use that these are homeomorphisms so x plus u you can write it as union of little x plus u they are copies of u therefore the each of them is open therefore x plus u is open similarly why u is union of alpha u where alpha runs over y alpha u is open because alpha is a alpha multiplication by alpha is a homeomorphism okay so we shall stop here and reap the harvest of all these observations if you have made next time remember the definition of convexity balance is that local convexity boundedness these are the new concepts I have introduced so I next time we will you know use these things and produce many interesting results here thank you