## Study Guide

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## Study Guide

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- School:
- Massachusetts Institute of Technology
- Course:
- 18 155 - Differential Analysis

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SIXTH ASSIGNMENT DUE OCTOBER 23 IN CLASS 18 155 FALL 2001 RICHARD MELROSE Problem 1 Hilbert space and the Riesz representation theorem If you need help with this it can be found in lots of places for instance 1 has a nice treatment i A pre Hilbert space is a vector space V over C with a positive definite sesquilinear inner product i e a function V V 3 v w 7 hv wi C satisfying hw vi hv wi ha1 v1 a2 v2 wi a1 hv1 wi a2 hv2 wi hv vi 0 hv vi 0 v 0 Prove Schwarz inequality that 1 1 hu vi hui 2 hvi 2 u v V Hint Reduce to the case hv vi 1 and then expand hu hu viv u hu vivi 0 ii Show that kvk hv vi1 2 is a norm and that it satisfies the parallelogram law 1 kv1 v2 k2 kv1 v2 k2 2kv1 k2 2kv2 k2 v1 v2 V iii Conversely suppose that V is a linear space over C with a norm which satisfies 1 Show that 4hv wi kv wk2 kv wk2 ikv iwk2 ikv iwk2 defines a pre Hilbert inner product which gives the original norm iv Let V be a Hilbert space so as in i but complete as well Let C V be a closed non empty convex subset meaning v w C v w 2 C Show that there exists a unique v C minimizing the norm i e such that kvk inf kwk w C Hint Use the parallelogram law to show that a norm minimizing sequence is Cauchy v Let u H C be a continuous linear functional on a Hilbert space so u Ck k H Show that N H u 0 is closed and that if v0 H has u v0 6 0 then each v H can be written uniquely in the form v cv0 w c C w N 1 2 RICHARD MELROSE vi With u as in v not the zero functional show that there exists a unique f H with u f 1 and hw f i 0 for all w N Hint Apply iv to C g V u g 1 vii Prove the Riesz Representation theorem that every continuous linear functional on a Hilbert space is of the form uf H 3 7 h f i for a unique f H Problem 2 Density of Cc Rn in Lp Rn i Recall in a few words why simple integrable functions are dense in L1 Rn R with respect to the norm kf kL1 Rn f x dx PN ii Show that simple functions j 1 cj Uj where the Uj are open and bounded are also dense in L1 Rn iii Show that if U is open and

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