By Paulo Ney de Souza, Jorge-Nuno Silva
This publication collects nearly 9 hundred difficulties that experience seemed at the initial assessments in Berkeley over the past 20 years. it really is a useful resource of difficulties and ideas. Readers who paintings via this booklet will boost challenge fixing abilities in such components as actual research, multivariable calculus, differential equations, metric areas, complicated research, algebra, and linear algebra.
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Extra info for Berkeley Problems in Mathematics (Problem Books in Mathematics)
Show that there is a continuous function T : L M,,, T ( A ) A= I , for all A, where I , is the identity o n R". 21 (Fa91) Let M,,, be the space of real n x n matrices. Regard it as a metric space with the distance function n d ( A ,B ) = laij - bijl ( A = ( a i j ) ,B = (bi,)) . i,j=l Prove that the set of nilpotent matrices in Mnxn is a closed set. 56 4. 1 (Fa93) Let X be a metric space and (x,) a convergent sequence in X with limit XO. } x ~ ,is compact. 2 (Sp79) Prove that every compact metric space has a countable dense subset.
Here, 11 . ) Assume 11 f(x)II < 11x11 f o r all nonzero x E B,. Let 20 be a nonzero point of B,, and define the sequence (xk) by setting xk = f ( z k - I ) . Prove that limxk = 0. 6 (Su78) Let N be a norm on the vector space Rn; that is, N : RT1-+ R satisfies N ( x ) 2 0 and N ( z ) = 0 only i f x = 0 , N ( . + 5 N ( x ) + N(Y)l N ( X x ) = IXlN(x) f o r all x,y E and X E EX. 1. Prove that N is bounded on the unit sphere. 2. Prove that N is continuous. 3. Prove that there exist constants A > 0 and B x E En,Alxl I N ( x ) 5 Blxl.
17 (Su77) Let f ( x ,t ) be a C' function such that = Suppose that J ( x , 0) > 0 f o r all x . Prove that f ( 5 ,t ) > 0 f o r all x and t. 18 (Fa77) Let f : IWrL tives and satisfy for all x = ( 5 1 , . . ) (where llull = = -+ R have continuous partial deriva- 1,.. , n. Prove that - f(Y)l I6KIIX - Yll duf+ . + u i ). 19 (Fa83,Sp87) Let f : R" \ ( 0 ) -+ R be a function which is continuously differentiable and whose partial derivatives are uniformly bounded: for all ( X I , . . ,x n ) # (0,.