Problem 1: Connectedness and Compactness in Topological Spaces Consider the topological space X = R² with the standard topology. Let A and B be subsets of X defined as follows: • • A is the open unit disk, i.e., A = {(x, y) = R² | x² + y² < 1}. B is the closed annulus, i.e., B = {(x, y) Є R² | 1 ≤ x² + y² ≤ 4}. 1. Prove that A is connected but not compact. 2. Prove that B is compact. 3. Let C AUB. Determine whether C is connected and justify your reasoning. 4. Draw the sets A, B, and C, and illustrate their topological properties using a Venn diagram or a similar graphical representation. Problem 2: Banach Spaces and Weak Convergence Let X be a Banach space, and let {n} be a sequence in X such that it converges weakly to some x = X, i.e., xnx. Assume that {n} is also bounded, meaning sup ||xn|| < ∞. 1. Prove that weak convergence does not necessarily imply strong convergence in a Banach space, by providing a specific example in the space IP (with 1 < p < ∞). 2. Show that if X is a Hilbert space, then a sequence {n} converges weakly to a if and only if (xn, y) (x, y) for all y € X, where (.,.) denotes the inner product. 3. Illustrate the distinction between weak and strong convergence with a graph showing the convergence of norms versus inner product convergence for different sequences.

If V1, V2, V3, V4, and W are all vector spaces.  Are L(V1x...xV4, W) and L(V1, W) x...x L(V4, W) isomorphic vector spaces?

1 find an o-thogonal bans for the space spanned by bac set {(0,2,1),,0,1), lo,4.0), (4,2,2)} d fi

2. Find the complement CC of the set C with respect to the space C if (a) C = {x :0 < x < 1}, C = {x : < x < 1}. (b) C = {(x, y, z) : x²+ y² + z² < 1}, C={(x, y, 2) : x² + y² + z² = 1}.

Let V be an inner product space. Then for x, y, z ∈V and c ∈F, the following statements are true. (a) <x, y + z>= <x,y>+<x,z> . (b) <x,cy>=c . (c) <x, 0>= <0, x>= 0. (d) <x,x> = 0 if and only if x = 0 . (e) <x, y>= <x, z>for all x ∈V, then y = z.

Show that if X and Y Hausdorff spaces, then so is the product space X × Y. are

Show the 3-dimensional ball B₁: ((..2) ER³ | x² + y² + 2² ≤1²) is a Jordan set and Vol(B₁)= Vol (B₁). r> 0.

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3. Prove that the connected components of a space X1,..., Xn are sets of the form E x x En where each E, is a connected component of Xj. ...

one example of a 4-dimensional vector space that is not R 4