functional analysis

functional analysis

Multiple Choice Questions 1) Which of the following is not a vector subspace of the vector space of polynomials with real coefficients 0) { p(x ) | p 13 ) = 0} b) All polynomials without constant term ( ) { pix > | pix ) = p ( 1-* ) , * EIRY d ) All polynomials with integer coefficients 2) Let V be a five dimensional vector space , and let S be a subset of V which is a basis for V. Then S a) must have at most five elements b) must be linearly dependent ( ) must have exactly five elements d) must have at least five elements 3 ) For what value of of is the set of vectors ) ( 1, 1 , 1 ), (1, 2, 0) , 12, 3, x )7 is linearly dependents 0 ) - 1 b ) 0 c ) 1 d) 2 4 ) If furs is linearly independent and if fly, wy is linearly dependent, then Z is in span zxys. a ) True by False5 ) Let May be the vector space of 2*2 matrices and let K[ " ] , and define the subspace C = {AE MIZ | KA = AK] What is the dimension of C ? 0) 1 b) 2 c) 3 d) 4 6 ) Letv- ( 2 , - 1 , 4, - 5 ) E IR". What is the value of livlly? a) 0 by 6 ( ) 12 d)-b 7) Let fox)= 2x3- 3x2-12x + 1. Clearly fe C[=13) . What is the value of Il foxall ? a) -3 6) 8 c) -19 d) 19 8) Let a= 2n4 3 . Clearly ( a ) E (". What is the value of 1/all? 58) Let E be a subspace of a normed space . We say E is complete if a) Every convergent sequence in E is Cauchy. by Every sequence in E has a Cauchy subsequence ( ) Every Cauchy sequence in E, converges in E. do Every sequence in E has a convergent subsequence which Converges in E. 10) Let X be a normed space and ES X. Consider the following statements 51 : E is compact $2: E is closed and bounded $3 . Every sequence in E has a subsequence converging in E. Which of the following is not always true? 0) 51 52 6) 31953 0) 52951 0)53431