 Welcome to the last chapter, point set topology course part 1. So, this will be topological groups and topological vector spaces. Interaction of algebraic operation of addition scalar multiplication etcetera with the Euclidean topology on the one hand and the important example that we had Van Aak spaces right. They are very important ok. What are the Van Aak one Van Aak space we can emphasize namely the set of all bounded functions on a given set right. So, all these things together motivate the study of the so called topological groups on the one hand and topological vector space is on the other ok. So, this last chapter is devoted to a brief introduction to these two concepts. The first section is just a brief introduction to topological groups and the next section we will have take up topological vector spaces ok. It is no way claims that this is exhaustive or comprehensive. This is just a brief introduction ok. Let me start with recalling the definition of what is a group first so that I can use those notations comfortably. I assume that you know already groups because we have even studied group actions right. So, a little bit of group theory whatever groups, homomorphisms, subgroups etcetera I suppose you know, but let me recall them first for ready reference also. First of all you may treat a group as a you know order triple of a set G and a binary operation mu and a distinguished element E where G is a set mu is G cross G to G that is the meaning of binary operation which is associative. If you have a short notation we will use mu of G comma H as G composite H. So, G and H are elements of G. G circ H is also inside G you may relate a G composite H or G circ H. Axiom of identity there is one axiom which means another another condition here is that this distinguished element is called identity why because its action on other elements E composite G is G and on the right side of the G composite is also G for every G H ok. So, it is acting identically on other element that is why the third axiom is that for every G in G there exists a unique element G inverse ok. G to the minus 1 we written as G inverse such that what is the property G composite G inverse equal to G inverse composite G is the identity ok. The element E is called the identity element which I have been referring to for every G in G the element G inverse is called actually inverse of G. We will also use the short expression G is a group just like this X is at a logical space. So, we can suggest that G is a group instead of writing the triple G mu E etcetera each time often even the simple notation is composite instead of mu of G H we are writing G circ H, but most often when there are no other compositions you will just write as G H. If there are two three compositions then we have to distinguish we cannot write all of them as G H ok. Even there are two different groups ok one is a domain and a core domain we are using the same G H here and G prime H where I am there to mean that we have to take the corresponding multiplications from G and G prime respectively. So, such short you know short notations or abuse of notation which is there in the practice by stalwarts all the way go back to you know Euler and so on we cannot change that so we better follow those rules. In a group G if the composition happens to have this property namely G composite H could H composite G for all G and H then G is called an Abelian group or a commutative group ok this is similar to the case of integers, rational numbers, real numbers and so on. So, there the standard notation is plus G plus right G plus to G composite H, but that is not forced on us because there may be more than one composition both of them commutative ok. So, then you have to choose which one is plus plus and plus prime and so on. So, that is also not it is only customary you have to follow the customs here rather than rigid rules. Rigid rules will be followed you know logically in our mind that much we have to do by abuse this abuse of notations it just means that you are not going to get confused by this notation that is all I did ok. Let G be a group together with a topology on the underlying set G such that the product map namely x comma y going to x y inverse that I am going to denote by nu for a y that must be continuous ok. What is the continuity here G is a topological group it means it has topology ok. So, it is a multiplication is here then condition is that this multiplication is continuous means what here there is a topology here I have to take the product topology ok. The product topology from given topology on both G and G here that is what I have to take under that this must be continuous ok. We then call G is a topological group a subgroup edge of G together with the subspace topology will be called a topological subgroup provided what what you have to do the group operation is also the same because it is a subgroup ok. So, however when the context is clear we may simply mention this as a subgroup likewise homomorphisms from G to H between topological groups ok are always assumed to be continuous unless mentioned otherwise ok. Precomposing the continuous function nu from G cross G to G by y going to e comma y see y going to e comma y is a group home is a continuous function from G to G cross G right it is like a coordinate inclusion followed by x y going to x y inverse e becomes x. So, this x becomes e sorry. So, this becomes just y going to y inverse. So, that map I am denoting by eta y it is called the inversion map eta y is y inverse because y inverse uniquely defined ok. This map is called what inversion map it is called it becomes continuous by very by this observation that is no I am not making this as a hypothesis now this is a follow this is a consequence because this is a composite of these two functions likewise if you take x y going to see first composite nu and then take eta. So, x y going to x y inverse is your identity comma eta right. Therefore, these are continuous now composite with x y inverse that will become x y going to x y. So, nu of x comma y inverse is x y inverse y inverse right. So, that is equal to x y. So, this is actually the multiplication map nu which you started with that will be also continuous. So, in one single go by taking this definition namely x y going to x y inverse we have made taking inverse taking inverse of inverse of an element also taking x going to x y both of them are continuous ok. Once you have both of them continuous you can recover the continuity of nu g inverse continuous nu net limit x y going to x y inverse also. So, they are equivalent instead of one you may have two different conditions combined with that one it will give you the first condition. So, they are equivalent ok. So, 32 and 33 together imply continuity of mu all right. So, here are some examples now any group together with the discrete topology or in discrete topology is a typical group. See you have already a group. So, group operation you do not have to change. You put a topology we are very familiar with putting lots of different topologies. You put a topology discrete topology cross discrete topologies discrete the codominious discrete, but any function from a discrete topology to any other space is always always what continues. Similarly, indiscreet indiscreet to indiscreet any function into an indiscreet space is also continued. So, out of this discrete topology is not so, it is interesting we have already done one of them namely action of a discrete discrete group on a set and so on. But the latter is namely indiscreet topology is most interesting one we will never have an occasion to use that one ok. But what may one what may you know trigger some thought process here is that these two extremities are there right. They are both topological groups without change a group. So, you you may think that topological groups after all have no special properties at all. A topology for a topological group may not have no extra properties at all. If at all there are properties it must be because of the group theory you may see wait a minute that is not the case. So, we are going to prove something out of this maybe you you may think it is nothing ok. So, not all topologies on a topological group even if you change the group structure maybe will be topological groups. So, we will see such things. So, that being the abstract part let us come back to some reality namely genuine and useful examples. The real numbers, complex numbers these were the motivating examples for them for us for this abstract definition right along with standard addition and standard you know if you take non-zero complex number there is multiplication also they are all topological groups right. The complex numbers of unit length they form a closed subgroup of the respective multiplication topological groups complex or real numbers ok. If take unit length of real number they are just minus 1 plus 1 that is some group. Similarly, the circle ok unit circle is a subgroup of the non-zero complex numbers under manipulation ok. So, these are the easy examples any finite dimension vector space over k is also a topological group. I am only looking at the addition the scalar multiplication is there we will study them little later. Similarly, our example namely I told you about Panak spaces the set of all bounded functions on x taking values in either r or c ok that was a Banak algebra right. So, there the addition will be automatically continuous with respect to the topology induced by the norm the norm is supremum norm ok. One of the most interesting case of all these topological groups occur inside the matrix matrices of you know n cross and matrices addition there already makes it a topological group ok. Even the multiplication just like the in the case of complex number central number of course you have to throw away the 0 the 0 matrix right. Not only 0 matrix you have to throw away this time you have to throw a lot more namely all matrices of determinant not equal to 0 determinant equal to 0 you have to throw away. So, in other words you have to take only invertible n cross n matrices invertibility with respect to multiplication. So, that will form a group and the group laws are continuous. How do you take inverse of a of an invertible matrix n cross n matrix? So, each entry will be a polynomial namely the ijth j ith cofactor divided by the determinant. So, it is polynomial divided by polynomial, but the denominator is non-zero therefore, they are continuous ok. So, glnk forms a what a group and the group laws are continuous with respect to what? With respect to the euclidean topology there glnk is an open subset of all k k you know n cross n I have written which I have told you earlier namely you write it as you know n cross n vector you think of that as n cross n vector. So, that way you get a euclidean topology on that ok. Similarly, you can look at o n which is defined as all those matrices real matrices ok. So, set A A transpose is identity. If you take complex matrices and then take A A star namely conjugate transpose right A A star that is equal to identity that will be called the unitary group. So, verification that their groups is easier it is linear algebra matrix theory. The only missing thing was why the group multiplications are continuous. You can separately verify that A comma B going to AB is continuous by looking at the matrix entries of this product they are all polynomials. Then you have to look at whenever this is invertible then how to write the inverse. So, for writing the inverse you have the Kramer's rule which says take the adjoint matrix which is again each entry is a polynomial in the original one. Then each entry you have to divide by the determinant of the given polynomial given matrix. So, that is also another polynomial right. So, that is the whole idea of this g l n c and g l n k in general g l l r and g l n c special cases. They have various subgroups I have denoted only two of them here o n and s u n here. Then you can take groups with determinant one also here. They will also form another subgroup smaller subgroup and so on. So, these are groups and many, many other groups are you know central to a lot of mathematics and there are theories here a small aspect of this you can call them as matrix groups which will lead to what are called as later on a deep theory very beautiful theory lead groups which we will not be able to do. So, let us stop here and take up these discussions next time. Thank you.