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Tag Archives: universal properties
Polynomials and Representations XXXVII
Notations and Recollections For a partition , one takes its Young diagram comprising of boxes. A filling is given by a function for some positive integer m. When m=d, we will require the filling to be bijective, i.e. T contains {1,…,d} and each element occurs exactly … Continue reading
Tensor Product and Linear Algebra
Tensor products can be rather intimidating for firsttimers, so we’ll start with the simplest case: that of vector spaces over a field K. Suppose V and W are finitedimensional vector spaces over K, with bases and respectively. Then the tensor product is the vector … Continue reading
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Tagged bilinear maps, duals, linear algebra, tensor algebra, tensor products, universal properties
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Topology: Quotients of Topological Groups
Topology for Coset Space This is really a continuation from the previous article. Let G be a topological group and H a subgroup of G. The collection of left cosets G/H is then given the quotient topology. This quotient space, however, satisfies an additional … Continue reading
Posted in Notes
Tagged advanced, group quotients, open maps, quotient topology, topological groups, topology, universal properties
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Topology: Quotient Topology and Gluing
In topology, there’s the concept of gluing points or subspaces together. For example, take the closed interval X = [0, 1] and glue the endpoints 0 and 1 together. Pictorially, we get: That looks like a circle, but to prove it’s … Continue reading
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Tagged advanced, gluing, klein bottle, mobius strip, quotient topology, topological groups, topology, universal properties
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Topology: Complete Metric Spaces
[ This article was updated on 8 Mar 13; the universal property is now in terms of Cauchycontinuous maps. ] On an intuitive level, a complete metric space is one where there are “no gaps”. Formally, we have: Definition. A … Continue reading
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Tagged advanced, cauchy sequences, complete metric spaces, completion, metric spaces, topology, universal properties
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Topology: Product Spaces (II)
The Box Topology Following an earlier article on products of two topological spaces, we’ll now talk about a product of possibly infinitely many topological spaces. Suppose is a collection of topological spaces indexed by I, and we wish to define … Continue reading
Intermediate Group Theory (6)
In this post, we’ll only focus on additive abelian groups. By additive, we mean the underlying group operation is denoted by +. The identity and inverse of x are denoted by 0 and –x respectively. Similarly, 2x+3y refers to x+x+y+y+y. Etc … Continue reading
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Tagged abelian groups, advanced, direct products, direct sums, free groups, generated groups, universal properties
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