Prove that an affine map is continuous if and only if its linear part is continuous

Q: (a) Define the T1-space.

A: Define the T1 - space

Q: › affine space (E‚È‚+), let (ai)ie is linearly independent for some i

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Q: Show that R is a Banach space.

A: Banach space: A complete normed linear space is called Banach space. Given: ℝ3=x(1, x(2), x(3)) :…

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: A topological space is Hausdorff iff limits of all nets in it are unique-

A: This is a problem of Topology.

Q: what is reflexive Banach space ?

A: Banach space: A normed linear space N,· is called banach space if it is complete metric space. A…

Q: be a closed subset of space Xa

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Q: The Joukowski map is defined by

A: Given, the Joukowski map is defined by                                     w=f(z)=12z+1zTo prove…

Q: Prove that linear maps are bounded for multivariable calculus

A: In this question, concept of linear map is applied. Linear Map The matrix of the linear map can…

Q: Let X be a locally compact Hausdorf f space

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Q: Prove that every well order is a linear order,

A: A set  is said to be well-ordered if it has a total order and has the property that every non-empty…

Q: Prove: Let X and Y be two topological spaces. Then the function f:X →Y is continuous if and only if…

A: This is a problem of topology.

Q: (c) Prove that the adjoint of a compact operator is compact.

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Q: Prove that coo cannot be a Banach space under any norm.

A: To prove: c00 cannot be a Banach space under any norm.

Q: Let X be a topological space

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Q: Prove that if X is a Banach space, X ∕= {0}, then there is a non-zero biounded linear functional on…

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Q: (a) Prove that the norm limit of a sequence of compact operators is compact.

A: Solution :-

Q: normed lineer spare X is finite demisional Linear transformation. X is bounded om If them every

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Q: How would you show g is a contraction map?

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Q: Exercise 4.2. Let X be a space with the finite complement topology. Show that X is T1.

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Q: Prove that linear maps are bounded.

A: To prove that linear maps are bounded.

Q: (a). Define three different metrices d1, d2, d3 and their induced norms on R³. Also prove the…

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Q: (c) Show that T-l is not bounded, and explain why this is not in contradiction with the Open Mapping…

A: Af first we will show that T is bijective and hence its inverse T -1 exists. Then we will find that…

Q: Is p continuous? Is p an open map?

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Q: how to prove a map is unique?

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Q: Determine the image of the line Im(z)= -2 under the mapping f(z)= i(z ) and describe it in words.

A: Im(z)=-2 represents a straight line y=-2let z=x-2i where x is real number. ⇒u+iv=f(z)=iz2=ix+2i2…

Q: Let X = R. Define an equivalence relation ~ on X by x ~ y if %3D and only if x – y is an integer.…

A: Let X, ?X be a topological space, and let ~ be an equivalence relation on X. The quotient set,…

Q: Prove that any reflexive normed space is complete

A: A normed space X will be said to be complete if each Cauchy sequence in X is convergent in X.

Q: Prove that any linear map between finite dimensional normed linear spaces is continuous.

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Q: The quotient topology actually defines a topology.

A: The quotient topology  actually defines a topology

Q: Provide an example of a topological space that is connected but not path-connected,why?

A: We use definition of connected and path connected sets. We use contradiction method.

Q: Prove: If F is a closed subset of a compact space X, then F is also compact.

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Q: Let X be a finite dimensional normed space (a) and A :X→X is a linear map. Show that A is compact.…

A: Given: X is a finite dimensional normed space and A : X → X is a linear map. We have to show that A…

Q: Prove that the image of a continuous mapping of a compact set is compact.

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Q: Prove that a linear map on a normed vector space is bounded if and only if it is continuous.

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Q: / Let (X₁T) and (Y₁J) be top. Spaces and fix-x be a surjective and continuous map, then y is Compact…

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Q: 2|A:k||2|| Bx:||2) < || |||B||F k%3D1

A: Please check the answer in next step

Q: Let p:Y ->X be a covering map. Prove that any fiber is homeomorphic to a set with a discrete…

A: Let f:X--&gt;Y be a local homeomorphism. Suppose y∈Y and let x∈Fy: = f-1[{y}], the fibre of y. Then…

Q: Suppose q: X → Y is a surjective map. Show that q is a quotient map if and only if the following…

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Q: Every continuous map T2 –→ S2 is homotopic to the constant map

A: To prove: Every continuous map from the Torus T2 to the sphere S2 is homotopic to the constant map.

Q: a discrete space

A: We know that in a discrete topological space X, every point in X is an open set.

Q: Let T be a linear operator on a finite-dimensional Prove that T is normal if and only if omplex…

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Q: Let N and N' be normed linear spaces with the

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Q: nach limits to show that the space ℓ∞ is not reflexive.

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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: Show that if a contraction map f: from M to M has a fixed point then it's unique.

A: Definition: Let X, d be a metric space. The map T: X→X is said to be a contraction mapping if there…

Q: (a) Let X and Y be normed spaces and F :X→Y be a continuous linear map. Show that F is closed . Is…

A: Given that,                   X and Y are normed spaces and F : X → Y is a continuous linear map. We…

Q: Show that a space X is Hausdorff if and only if the diagonal {(x, x)|x € X} is closed in X2.

A: We use basic topological definition to prove this.

Q: Define A : L3/2[–1, 1] → C by Prove that A is a bounded linear map and compute its norm.

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Q: A function is invertible if and only if it is a bijection. True or False?

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