Unformatted text preview: d 1 ( m,n ) = d ( m,n ) / [1 + d ( m,n )] . 3. Let M denote the set of continuous functions f : [0 , 1] → R on the interval [0 , 1] . Show that d ( f,g ) = Z 1  f ( x )g ( x )  dx is a metric. 4. Let E = F = R n with the standard norms and A,B ∈ L ( E ; F ) . Let < A,B > = trace ( AB T ) . Show that this choice is an inner product on L ( E ; F ) . 5. Let t = 2 e 1 ⊗ e 1e 2 ⊗ e 1 + 3 e 1 ⊗ e 2 and φ ∈ L ( R 2 ; R 2 ) , ψ ∈ L ( R 3 ; R 2 ) be given by the matrices ± 2 11 1 ² , ± 0 11 1 0 2 ² . Compute: trace( t ), φ * t , ψ * t , trace ( φ * t ) , trace ( ψ * t ) , φ * t . 1...
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 Winter '10
 Morgensesn
 Topology, Metric space, new metric d1, Control Midterm Exam

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