Suppose that (X, |..|/x) and (Y, ||:y) are Banach spaces, and T : XxY – C...
where C is a Banach
spaces C^k[a, b]
Prove that Va e Cº, 3 [0, 1].
Let X,Y be topological spaces, and f:X->Y a homeomorphism, I.e. f is one-to-one, onto, and f and f-1 are continuous. a.) Prove that if X is T4, so is Y. b.) Prove that if X is separable, then so is Y.
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous,
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous,
4. Let A, X, Y, Z be normed vector spaces and B :X Y + Z be a bilinear map and f: A+X,9: A + Y be mappings that are differentiable at to E A. Show that the mapping 0 : A → Z, X HB(f(x), g(x)) is differentiable at Do and that dº(30)[h] = B(df (o)[N), 9(30))+ (f(x0), dg(xo)[h]) (he A).
real analysis
hint
9 Let co , a, and 〈æ be the Banach spaces consisting of all complex sequences x={ i-1, 2, 3,..., defined as follows: X E if and only if II x11 if and only if lxsup lloo. for which ξί (a) If y = {nJ E 11 and Ax = Σ ζίηǐ for every x ε co, then Λ is a bounded linear functional on (More precisely, these two spaces are not equal; the preceding statement exhibits...
2. Let (X, dx), (Y, dy) be two metric spaces, and f:X + Y a map. (a) Define what it means for the map f to be continuous at a point x E X. (b) Suppose W X is compact. Prove that then f(W) CY is compact.
3. (1 point) Let (X.11 . ID be a Banach space. K C X be a closed subset and Assume that D40. Prove that the above equality holds true if and only if
3. (1 point) Let (X.11 . ID be a Banach space. K C X be a closed subset and Assume that D40. Prove that the above equality holds true if and only if
We used definition of homeomorphic as follows.
If X and Y are topological spaces, a function f: X to Y is
called homeomorphism if
1. f is continuous
2. f is bijective
3. inverse of f is continuous
And in this case, we say that X is homeomorphic with Y.
Thank you !
infinite) (5) Prove that all semiopen intervals in R (finite or homeomorphic are
infinite) (5) Prove that all semiopen intervals in R (finite or homeomorphic are
2. Let (X, dx), (Y, dy), (2, dz) be metric spaces, and f : XY,g:Y + Z continu- ous maps. (a) Prove that the composition go f is continuous. (b) Prove that if W X is connected, then f(W) CY is connected.
9. Let X and Y be metric spaces, and let D be a dense subset of X. (For the definition of "dense, see Problem 4 at the end of Section 3.5.) (a) Let f : X → Y and g : X → Y be continuous functions. Suppose that f(d)gld) for all d E D. Prove that f and g are the same function.