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.

Q: Consider V = {(2,y) : x, y ≤ R}. Let and ⋆ be operations on this set defined in the following way:…

A:

Q: If a CW-complex X has one 0-cell, two 1- cell, one 2-cell, and no other cells, then the space X is…

A: We need to determine the below statement is true or false. Statement is : If a CW-complex X has one…

Q: Elaborate on the role of duality in topological spaces.

A: Introduction:Topology is a branch of mathematics that studies the properties of spaces and the…

Q: Draw the relation

A: Given: x,y/x2+y2≤1 To draw: The relation.

Q: (b) Let {Ai e I}. Prove that 4) =UA;. |A, iel

A: We need to prove that ∩i∈IAic=∪i∈IAic  for set Ai/i∈I We will prove using the following method: To…

Q: Show that X is Hausdorff if and only if the diagonal △ = {(x, x)|x ∈ X} is closed in X × X.

A:

Q: Consider the set V = {(x, y) = R² : x+y=1}. on this set defined in the following way: (x2, y2) = (x₁…

A:

Q: Dijec not need to prove you function is a bijection): (a) N and Z (b) Rand (1, ∞) (c) (0, 2) and (3,…

A:

Q: Suppose both f(z) and the conjugate of f(z) are holomorphic on a connected open set.  Show that f is…

A:

Q: Show that R is not compact.

A:

Q: get the following sets {(x, y) : x ∈[1, 2], y ∈[1, 2]} and {(x, y) : x = 2, y ∈[0, 1]}

A:

Q: Let X = {1, (b) xm} and Y = {y₁, Yn} be finite sets. Determine the number of one to one functions…

A:

Q: 2. Determine whether the set R2 with the operations (x1, y₁) + (x2, Y2) = (x1x2, 9192) and c(x₁, y)…

A:

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: Let v be the set o all pairs of positive numbers (X,y) With two operations + and C denned as: (x1,…

A:

Q: 4 -1 33. The dimension of the nul-space of A = 3 3 is 2 1 1 2 3 (A) 1 (B) 2 (C) 3 (D) 0 (E) 4 142 13

A: Given matrix A=5214−14332211123 We have to find the dimension of null space. For dimension of null…

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

A: