By Robert J. Zimmer

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Let A E GL(n,R) and TA: C°°(R") —+ (a) Show that TA induces a bounded operator on LPk(Rn). (b) Show that TA induces an isometry of L2*(Rhs) (with respect to some equivalent norm) if and only if A is orthogonal. 18. Show Cr(R") is dense in but not the norm topology. 19. (a) Suppose E,F are Banach spaces and Ac B(E,F). 4) is bounded in R. pose that for all x E E, (IfTxjf T Topological Vector Spaces and Operators 37 Show (11Th) is bounded. Hint: Let AN = {x E h)TxhI S N for all T E A). Then use the Baire category theorem.

PRooF: First suppose K is of the form K(x, y) = for E C(X). ,'pn). In particular, Tx has finite rank and hence is compact. For a general K, we can find K1 of the above form with K, K uniformly. ) Since K, —+ K uniformly, K, K in L2(X x X), and hence — 0. 3. 5, assuming only that K L2(X x X). 6. Let E be a Hubert space with orthonormal ba- If T E B(E), then T is called a Hubert-Schmidt operator sis if 1T1112 <00, <00. e. Equivalently While this appears to depend on the basis, we show it does not.

F One of the most useful features of the weak-i-topology is the following result. 28. Let E be a normed linear space. Then (the unit ball in E) is compact with the PROOF: The proof of this will follow easily from Tychonoff's theorem: the product of compact spaces is compact. Namely, for each is compact, and = {c E k z E E, let lixill. Then hence so is = IIZEEBX, with the product topology. There is a natural map i: E £2, namely (i(A)), = A(x). e. 1. Since a net Wa £2 converges to w E £2 if and only if (W0)t Wr for all z, it follows It therefore sufhomeomorphism of with i(E7).