Olet se be u lecally compact topological space. Prove that that GL2) in a cEalyehra (ie. prove (a) Colse) ü a normed *-alyehrn. ( b) Corse) es Complete. (C) lt *Fl = 1F1" ( the cEalyebraic arrom ).

Q: Use Hahn-Banach Theorem to show that a normed linear space X is finite dimensional if its dual X' is…

A:

Q: u* (B) = inf{l&l:s is a covering of B by closed boxes} for all BcR" صواب İhi

A:

Q: Let the ordeced two bases of the spaces B be A=- (1,3) (E),L)} and B=fF3,2).(5,1)} ln the case, find…

A: We required to find the transition matrix from base B to base A Which is given follow

Q: if a family iP; Sierof sominorms on avectorspace X separates distinct points of x , show that the…

A:

Q: Suppose Y=[ 0 ,2 ) . In this case A=[1,2 ) is not closed in X; while closed in Y. In addition, G=[0…

A:

Q: Find a condition on a, b, c so that (a, b, c)  ∈ R^3, belongs to the space spanned by →u = (2, 1,…

A:

Q: Prove that any Space is complete. finite dimensional normed

A: Introduction: Completeness is a property of a topological space. When in a topological space, every…

Q: Define diffeomorphish between two manifolds and Completly explain why any linear map fiM_N. between…

A:

Q: Let V, W be finite-dimensional inner product spaces and T € L(V, W). Prove: (a) If T is injective,…

A: Injective Function:  A function is said to be injective if each element of the codomain is mapped to…

Q: Let X be a locally compact Hausdorf f space

A:

Q: There are several norm sok prac tical importante on the vector space of ordlered s n-tuplesLnotably…

A: A norm is a real-valued function defined on the vector space that is commonly denoted {\displaystyle…

Q: Q3/(a) Let d be a trivial metric on a set of X, find the induced topology on X by a metric d. (b)…

A:

Q: * Every Pre-Hilbert space is I - Continues Normed space -orthogonal Metric space inner Produced

A: Every pre-Hilbert Space is a complex vector space X together with an inner product (A complex inner…

Q: Give an example of a closed subset of R that is not sequentially compact -Give an example of a…

A:

Q: If X is vector space and (X,d) is discrete metric space, then X can induce an inner product space…

A:

Q: Exercise 5. (Let X,Y be two connected topological spaces. Consider AC X and nd BCY} 1) Assume that…

A: Given: The spaces X,Y are connected topological spaces. The non-empty sets A and B are the subsets…

Q: Q7) (a) Let X be a Banach space and R the usual normed space. Show that ||(z, a)|| = ||||+|al is a…

A: (a) Given a Banach space  and  the usual normed space, we want to show that the norm  is complete on…

Q: Suppose H is an inner |(x,y)| ≤ 11x11-1111 product space. Show Ужуєн.

A: Given: H is an inner product space. To prove: x,y≤x·y, ∀x,y∈H

Q: 20. Prove that a vector u in a vector space has only ative, -u. one neg-

A: By definition of negative of an element in a vector space, we have        u + (-u) = (-u) + u = 0 .…

Q: Let (X, T) be a topological space and A, B be a subsets of X then: اختر احدى الدجابات int(A - B)=…

A:

Q: По The finite closed interval [a, b] is compact and no in both the Standard cofinito topology.

A: Soln: The interval [a,b] being closed and bounded in R, thus [a,b] is compact in the standard…

Q: With the help of exomples Show thaat on finite dimensional are equivalent. any two norms Linear…

A: Given: Any two norms on finite dimensional linear space are equivalent.

Q: O Let ( X, T) be topdogical space and A,B CX- ldentify the Condition when a (Int(A)u FrlB) Ul F CA)…

A:

Q: 9. For each n E IN, let Xn be the set of polynomials of degree equal to n, and let Pn = U_0 Xk. True…

A:

Q: 12 Let (RTR) is topological Space Such that TR-243 U2GSR=R-G-G is finste} Find buz), Z², Z. z' then…

A: Here given ℝ, Tℝ is a topological space such that Tℝ=ϕ∪G⊂ℝ: ℝ-G=Gc is finite This topology is known…

Q: Letxandy and z be normed space s and s: xy andTIYz bounded oper atorsprove

A:

Q: 2. Determine which spaces under the given operatons. a)the set of all real- valued functions f…

A:

Q: 3) Find the dimension of the vectora space ganenated by the yeetons: '= (o 1 82= (2 -1 5 4) 83- (4 0…

A:

Q: Let T_us be the usual topology on R and T_|| be the lower limit topology generated by the unions of…

A: A homeomorphism is a bijective continuous map whose inverse is also continuous

Q: Lot X= [a,b] with the standard topology and let Y= [9₁6) with the Cofinito topology. Why is the…

A:

Q: (йх'ў):(wx2)

A:

Q: Let T_us be the usual topology on R and T Il be the lower limit topology generated by the unions of…

A: Homeomorphism is a bijective continuous map whose inverse is also continuous

Q: thun no in

A:

Q: Let (M, d) be a metric space and K ⊂ M. The following statements are equivalent: (i) K is relatively…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Q2. Answer as true or false: (a) f-'((AUB)nC) = (ƒ-'(A)n f-1C)) U (f-1(B) n f-1(C)). (b) Given the…

A: (a) is True. (b) is False.

Q: Prove by sequences that the open balls in R^n with the usual (Euclidean) metric are not compact

A: In this question, we have to show that open balls in ℝn with the usual metric are not compact by the…

Q: Prove the locally compact normal linear space is finite dimensions.

A: We have to prove the  locally compact normal linear space is finite dimensions First we define the…

Q: Let T us be the usual topology on R and T_Il be the lower limit topology generated by the unions of…

A:

Q: Determine whether the set equipped with the given operations is a vector space. For those that are…

A: Given set of ordered pairs of the form (7,x). To verify t is a vector space under the given…

Q: Exercise 3. In this exercise (X, d) is a metric space. .} 2) Assume there exists r> 0 and a sequence…

A:

Q: quutiont of a compact space is compact and connected Show that connected

A: Show That :  The quotient of a compact (connected) space is compact (connected).

Q: 13. Let X be a set, let (Y, Ty) be a topological space, and let f: X→Y be a function between sets.…

A: We shall show that tau is a topology in the given space. Please check the next step for details.