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1. Prove that any two norms on Rn are equivalent. (HINT: You may want to use the fact that the unit sphere with respect to the Euclidean norm in Rn is compact.) 2. Let M be a closed subspace of a Hilbert space x and f € M'. Show that there is a unique F € X* such that FM = f and IFII = II/II- Show that this extension vanishes on M- 3. Let {va|aEA be an orthonormal list in a Hilbert space X. Show that {ua}. is complete if and only if for all I.Y E X. (Be very careful.) 4. Let x be a Banach space and [/2} C X*. Suppose that for each I € X, f(x) = lim, fo(x) exists and is finite. Show that f €X*. 6. Let x be a Hilbert space. Show that if X* has a countable dense subset, so does X.

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Functional Analysis - Hilbert And Banach Spaces
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