quotient - FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT...

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Unformatted text preview: FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied the basic notions of abstract algebra, the concept of a coset will be familiar to you. However, even if you have not studied abstract algebra, the idea of a coset in a vector space is very natural: it is just a translate of a subspace. Example 1.1 (Cosets in R 2 ) . Consider the vector space X = R 2 . Let M be any one- dimensional subspace of R 2 , i.e., M is a line in R 2 through the origin. A coset of M is a rigid translate of M by a vector in R 2 . For concreteness, let us consider the case where M is the x 1-axis in R 2 , i.e., M = { ( x 1 , 0) : x 1 R } . Then given a vector y = ( y 1 ,y 2 ) R 2 , the coset y + M is the set y + M = { y + m : m M } = { ( y 1 + x 1 ,y 2 + 0) : x 1 R } = { ( x 1 ,y 2 ) : x 1 R } , which is the horizontal line at height y 2 . This is not a subspace of R 2 , but it is a rigid translate of the x 1-axis. Note that there are infinitely many different choices of y that give the same coset. Furthermore, we have the following facts for this particular setting. (a) Two cosets of M are either identical or entirely disjoint. (b) The union of the cosets is all of R 2 . (c) The set of distinct cosets is a partition of R 2 . The preceding example is entirely typical. Definition 1.2 (Cosets) . Let M be a subspace of a vector space X . Then the cosets of M are the sets f + M = { f + m : m M } , f X. Exercise 1.3. Let X be a vector space, and let M be a subspace of X . Given f , g M , define f g if f g M . Prove the following. (a) is an equivalence relation on X . c circlecopyrt 2007 by Christopher Heil. 1 2 QUOTIENT SPACES (b) The equivalence class of f under the relation is [ f ] = f + M . (c) If f , g M then either f + M = g + M or ( f + M ) ( g + M ) = . (d) f + M = g + M if and only if f g M . (e) f + M = M if and only if f M . (f) If f X and m M then f + M = f + m + M . (g) The set of distinct cosets of M is a partition of X . Definition 1.4 (Quotient Space) . If M is a subspace of a vector space X , then the quotient space X/M is X/M = { f + M : f X } . Since two cosets of M are either identical or disjoint, the quotient space X/M is the set of all the distinct cosets of M . Example 1.5. Again let M = { ( x 1 , 0) : x 1 R } be the x 1-axis in R 2 . Then, by Example 1.1, we have that R 2 /M = { y + M : y R 2 } = { ( x 1 , 0) + M : x 1 R } , i.e., R 2 /M is the set of all horizontal lines in R 2 . Note that R 2 /M is in 1-1 correspondence with the set of distinct heights, i.e., there is a natural bijection of R 2 /M onto R . This is a special case of a more general fact that we will explore....
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quotient - FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT...

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