7. Let a, b ER, with a < b. Prove that each discrete subspace of R and each of the spaces [a, b]. (a, b). [a,b). (a,b]. (-∞, a). (-∞, a]. (a, ∞), [a, ∞), and {a}, with the topology induced as a subspace of R, is a Polish Space.

Q: Q2) Let (A,T)be topological spaces; i=1,2,3. Then A, x A2 x Az is homomorphic to (A, x A2) x A3 I

A: Definition : Let X and Y be spaces, and let B be the collection of sets U × V , where U ⊆ X and V ⊆…

Q: 2) Let V₁, V2, W be vector spaces over F. Show that the set Bil(V₁ × V₂, W) of bilinear maps is a…

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Q: 4. Check that the following are vector spaces: (a) RN. 55 / 600 (b) CN. (c) C(N), where 2 is a…

A: A non-empty set V, +, · is said to be a vector space over a field F together with operations of…

Q: a. Is W = {ann + an-1xn-1 + •+aix+ao anan-1a₁a0 = 0} a vector space with the same operations in…

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Q: 3. Consider the set C(R) = {continuous functions of real numbers and consider its subsets. Show that…

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Q: 4. Determine whether M22 = { 1: a, be R} : a, be R M22 is not a vector space

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Q: Q3 Check the symmetry and graph the curve in polar form if 0 = 0 and equal 1/6 to 1/2 at increments…

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Q: f e 17. Show that the set of all points in R2 lying on a line is a vector space with respect to the…

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Q: Suppose X and Y are normed spaces and X + 0. Prove that BL(X, Y) is a Banach space with the operator…

A: Given, X and Y are normed spaces and X ≠ 0. We have to prove that BLX , Y is a Banach space with the…

Q: Q4. Consider the topological space (X,T), where X = {a, b, c, d) and T = {o, X, {a}, {b}, {c},…

A: Given: Q4. a)  X, T is a topological space, where X=a, b, c, d and T=∅, X, a, b, c, {a, b}, {a, c},…

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: ow that if (V, O, •) is a vector space then: = 0 for every u in V. (Use the definition of the zero…

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Q: 8. Suppose I is a triangle formed by placing three points lircle, two of which lie on the circle's…

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Q: 1 = { x:= = < |x|<1} am an 1. Consider the set Y=[-1,1] as a subspace of R. Is A=< open set in Y? Is…

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

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Q: The region bounded by f(x) 2x² + 12x + 14, x = = 0 is rotated about the y-axis. Find the volume of…

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Q: Q1) Prove or Disprove: Let L(F) be a linear space and A,B are two convex subsets of L. Then, AXB is…

A: The problem is related with convex set. Given that,                 LF  is a linear space  and A, B…

Q: Suppose that W1, W2,. , Wm and V are all finite dimensional vector spaces over F. Prove that 2,...…

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Q: (b) Suppose -|, and ||-||2 are two equivalen norms on a vector space X. If (X, ||.||,) is complete,…

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Q: Let A = {(1, –2,0), (0,0, 1), (-1, 2,0)}. Determine if A spans R. If A does not span R°, give a…

A: To find wheather A spans R3 we have to check if it forms a basis in R3

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Q: If W, and W2 are finite - dimensional subspaces of a vector space V, then W + W2 is finite –…

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Q: 18. Let V = {(a1, a2): a1, a2 € R}. For (a₁, a2), (b₁,b₂2) € V and c € R, define (a1, a2) + (b1,b2)…

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Q: 4. Let W be a subspace of a vector space V. We define a relation: V₁ V₂ if V₁ V₂ € W. ~ 4a. Show…

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Q: Q1. Let V be a vector space over a field (F, +, .), let S={ vi, v2, V3 } be a linearly independent…

A: Let V be vector space over a field (F, +, .).

Q: Prove that (F,|.|) is Banach space where FE{C, R}

A: Definition required:  Let X be a metric space and let xn be a sequence of points in X. (a) Cauchy…