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Unformatted text preview: B U Department of Mathematics Spring 2006 Math 332 - Real Analysis II, First Midterm, 17/4/2006, 17:00-19:30 Whatever theorem, proposition, lemma you use, Full Name : you must be sure that it is applicable. Student ID : Justify all your claims in your proofs. over 40 1.  For a function f : A ⊂ R m → B ⊂ R n and x ∈ A , fill in the blanks as precise as possible : Df ( x ) : R m → R n Df : A → L ( R m , R n ) D : set of diffble fncs → fncs from A to L ( R m , R n ) D 2 f ( x ) : R m × R m → R n D 2 f : A → bilinear fncs from R m × R m to R n 2.  Let I be the n × n identity matrix. What do we mean when we write Df ( x ) ´ = I ? Solution: LHS is a linear transformation while RHS is the corresponding matrix in standard bases. 3. Claim: Any continuous function f : [0 , 1] → [0 , 1] can be approximated by a contraction. (a)  Why cannot we use the Stone-Weierstrass theorem to prove this claim?...
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This note was uploaded on 05/04/2011 for the course MATH 321 taught by Professor Talinbudak during the Spring '11 term at BU.
- Spring '11