Task 8 Let (X, || · ||x), (Y, || · ||Y) be Banach spaces. Show that (L(X,Y), || - ||cx,.Y) is a Banach space.

Q: Task 8 Let (X, | | |x). (Y, |- |y) be Banach spaces. Show that (C(X,Y). lecx.x) is a Banach space.

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Q: (a) Define the T1-space.

A: Define the T1 - space

Q: Let B={(1,-2,0,-1), (0,1,0,3), (1,2,1.0)], and x=(4,3,-2,-5) Find [x]B Give your answer in the form…

A: B=1, -2, 0, -1, 0, 1, 0, 3, 1, 2, 1, 0x=4, 3, -2, -5

Q: 3A Given the Euclidean inner product space (R³(R),+,,*), where for each_x=(x1,X2,X3),…

A: The definition of inner product is given. We have to find the matrix representation of the inner…

Q: i: Let X and Y be topological spaces. Show that X Y X -Y but X -Y X - Y.

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Q: Let B={(1,-2,1), (4,-7,5), (5,-8,8)], and x=(-6,10,-7) Find [x]B

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Q: 3. is Show R with explamtion. Banach space, a

A: Banach space: The complete normed linear space is called Banach space. Given: To show: is Banach…

Q: Thm. Let v be an inner product Space then and HyE V жує ||≤|12||· 1191/ further equility is hold iff…

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Q: (a) Let M = R. Give the radius r and the center c of B(-2,5) n B(6, 7). ne

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Let v1 =[1,4,-2] , v2 =[-2,-3,7] , b =[4,1,h] FIND all value(s) h such that b is in span{v1 , v2 }

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Q: Let V = span{e2", xe2", x²e?=}. (a) Show that (a1e2a + a2xe2" + a3x²e2") e V for any a1, a2, az E R,…

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Q: Give example T2 space but not T3 Space

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Q: Let U = ((1, 1, 0, 1), (0, 1, 2, 3), (2, 3, 2,−1)) and W = ((1, 3, 4, 4), (1, 1, 0, 2)) be sub-…

A: The given U=1, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, -1 and  W=1, 3, 4, 4, 1, 1, 0, 2 be a subspaces of ℝ4.…

Q: Give an example of a function f : (X1, d1) → (X2, d2) that sends compact sets to compact sets, but…

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Q: if (X,d) is discrete metric space, ACX and A+D (x,d) Find the sets of A° and A (functional analysis…

A: Consider a discrete metric space X,d where d is defined as: dp,q=0,p=q1,p≠q

Q: Let (X, t) be a topological space and A, B EX, then 1- (X- A)° CX-Ā 2- (X-A)° 2X-Ā 3- (X- A)° = X-A…

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Q: Let (X, T) be a topological Space and ACX- Develop the relation between Fr ( Fr(A)) and Fr(A).

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Q: Q # 3(b) Let (G, H) be a disconnection of a topological space (X, 7), ant let AC X. If A is…

A: Before going to this proof, we prove the below statement which in turn proves the above result.…

Q: 4={(-21.r):reR} II) B=(2r.-31): te R} III) C={(2r+3,-41) re R} I)

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Q: Show that if a Banach space X is non-trivial, X + {0}, 3). then X' # {0}.

A: According to the given information, it is required to show that:

Q: Q4) Consider the space (X.Tu) and let A = [0,2) U (3) U (4,7). Find d(A), A, Aº, ext(A) and b(A).

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Q: Can you show an example of Null space?

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Q: Quertion:- Show that R is a Banach Space.

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Q: Task 8 Let (X, || ||x), (Y, ||· ||y) be Banach spaces. Show that is a Banach space. (L(X,Y). ||…

A: Given: X, ·X, Y, ·Y are Banach spaces. To show: LX, Y, ·LX, Y is a Banach space.

Q: R and C with ||x|| = |x| are Banach spaces.

A: Given: ℝ and ℂ with x=x.  We have to prove that they are Banach spaces.

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

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Q: ACX A + Ø and A + X. Let B = AU fr(A). .show that for the topology T = {X,Ø, A, B} it happens that…

A: Given A⊂X where X is non-empty and A≠X Suppose, T=X,∅,A,B be a topology.  Here B=A∪frA To prove for…

Q: Show R³ is a banach space.

A: Banach space: A complete normed linear space is called Banach space. Given: To show:  is a Banach…

Q: (c) If X={1, 2, 3}. T= {4, X, {1}, {1,2}, {3}, {1,3}}. JUSTIFY whether the topological space (X, t)…

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Q: B. Is C a real vector space? Explain.

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Q: Banach space. Show that X E and only if X' is reflexive.

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Q: Show that (C(X),|| · ||∞,x) is a metric space.

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Q: (b) space and X+{ 0 }. Show that B< (X, Y) is a Banach space. Let X be a normed space and Y be a…

A: X be a normed space and Y be a Banach space We have to show that B(X, Y) - set of all boundes linear…

Q: Every T2 (Hausdor ff) space is T1 space. True O False O

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Q: Let X = {a,b,c} and Y = {1,2,3,4}. Which of the relations A,B,C defined below are functions from X…

A: The given problem is related with function. Given that ,               X = a , b , c and Y = 1 , 2 ,…