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1 let s d be a complete separable metric space with s non-empty sup- pose s has no isolated points that is for each x
1 for a finite measure space x s micro suppose there are points xi and numbers ti gt 0 such that for any a isin s
1 in any infinite-dimensional f -space show that any compact set has empty interior2 a continuous linear operator t
3 prove that in the open mapping theorem 652 it is enough to assume rather than t onto that the range of t is of
1 show that for any two normed linear spaces x imiddoti and y middot and lx y the set of all bounded linear
recall that a radon measure on r is a function micro into r defined on all bounded borel sets and countably additive on
1 let f x x p for all real x for what values of p is it true that a f is convex b the derivative f ix exists for
1 for any set f and point x in a metric space recall that dx f infdx y y isin f let f be a closed
1 show that in any finite dimensional banach space rk with any norm for any closed convex set c and any point x not in
1 give an example of a banach space x a closed convex set c in x and a point u isin x which does not have a unique
1 give an example of a set a in r2 which is radial at every point of the interval x 0 x 1 but such that this
1 continuation if g is a function from x into s then t is called the pettis integral of g if for all u isin st ug
let x s micro be a measure space and s 1middot1 a separable banach space a function f from x into s will be called
let f be a lipschitzian function on x and e a subset of x where f l on x is the same as for its restriction to e
the support of a real-valued function f on a topological space is the closure of x f x 0 let l be the set of
1 let micro and nu be finite measures such that nu is absolutely continuous with respect to micro and f dnudmicro show
1 take the cantor function g defined in the proof of proposition 421 as a nondecreasing function it defines a measure
1 if micro and nu are finite measures on a sigma-algebra s show that nu is absolutely continuous with respect to micro
1 let fnk x ank for 2k le 2n x lt gt2k 1 and fnk x bnk for 2k 1 le x 2k 2 for each n 0 1 and k 0 1 k
1nbsp for any set f and point x in a metric space recall that dx f infdx y y isin f let f be a
1 suppose that a real-valued function f on an open interval j in r has a second derivative f ii on j show that f is