3.1 Let S = (1,2,3) and be a topology on S defined by r = {0, {1}, {2), (2,3), S). Then T (a) (S, T) is a normal space, (b) (S, T) is a regular space, (c) (S, T) is not a T₁-space, (d) {1) and (2, 3) are separated sets, 3.2 The set Q of rational numbers as subset of the set R of real numbers with discrete topology is open.

Q: (4) Suppose V and W are finite dimensional vector spaces such that dim(V) > dim(W). Show that any…

A:

Q: Solve the following oblique triangle using the Law of Sines. Show your work! Round 1 place after the…

A: We are provided with following information : B=47.2o , C=73.7o b=56.9 ft According to Sine rule , we…

Q: C) If = R,P₁ = {(-∞, -3], (-4,6), [6, 0)), B₂ = (Q.Z, R - Z) and 3 = {(a, b]la < b, a, b E R}.…

A: Answer given below.

Q: 1.3.1 Determine whether or not the given vector spaces V and W are isomorphic. *(a) V=R, W= M2,3(R)…

A: The two vector spaces are isomorphic if there exists a bijection between them T:U→V such that…

Q: Show that B= (u1, u2, u3) is linearly independent set in R. Show that B = (u1, u2, U3) is a spanning…

A:

Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X,M) := {f €…

A:

Q: 5- Solve the folowing linear algebraie equations using the Gauss elimination method: 3x -2x2 + 7x3 =…

A:

Q: 2. Let a, B, y be three vectors in the inner product space V½ (R) with the standard inner product…

A:

Q: Let S = {1, x, x²} be a subset of the vector space P(C) equipped with the inner product 1) = [₁, f…

A:

Q: or not. (c) Let f be a map from a space(X, ). into a space (Y,T) and t,T such that TSt and TS,, show…

A:

Q: 6. Let V be a vector space over F and let u, v, we V, a, Be F. Which of the following is not a…

A: In this question, we will discuss about the properties of vector space.

Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f e…

A:

Q: Q4) Let C(R) be a normed space such that || f -sup{ Ifl,x E R}and T: C (R) -R define Tf)-If I. Is T…

A:

Q: (r, x) = 0 implies z = 0; • (r, x) 20 for all z e H; • (r, y) = (y, x); • for any a € C, %3D (ar +…

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: For each of the examples below find an operator T* on a vector space space V such that for all x, y…

A:

Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

A:

Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and ВС (Х, М) :%3D…

A:

Q: Which of the listed mappings is an isomorphism of vector spaces?

A: (a) A:R3→R3​[x],A:(a,b,c)↦ax+bx2+cx3 Linearity:…

Q: d T = {0, {a}, {a, b}, {c,d}, {a, c, d}, {a,b, c, d}, X}. (1) Is this space To? Why or why not? (2)…

A:

Q: (v) Prove that if the space (X,T) has the fixed-point property and (Y, T1) is a space homeomorphic…

A: Given: v) To prove the given statement as follows,

Q: Let S = { (0,0), (0,3), (1,0), (1,2), (2,0), (3,2) } Find S', the transitive closure of S

A:

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

A:

Q: Is not closed set in X x X. Then the space X is: (a) Hausdorff Space (b) Not Hausdorff Space (c)…

A:

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: 20. Let V be an inner product space. Prove that (a) |||x|| - ||y||| ≤ ||x − y||. (b) (x, y) =…

A:

Q: Show the span {(1,0, 1), (-1, 2, 3), (0, 1, – 1)} is all of R'.

A: Let x=1, 0, 1, y=-1, 2, 3 and z=0, 1, -1. We have to show that spanx, y, z is actually the vector…

Q: Use the inner product (f,9) = f(z)g(x) dz in the vector space C"0, 1] of continuous functions on the…

A: It is given that fx=-5x2-3 and gx=9x-7. We have to evaluate f, g, f, g and αf, g where αf, g is the…

Q: 3. (a) Let S = {v1, . . . , vn} be a linearly dependent subset of a vector space V , and let vn+1 be…

A: This question is related to linear algebra.

Q: 1) The space (R²,O,O) is not a vector space where: (x, y) O (z, w) = (x + z + 1, y + w + 1) a O (x,…

A: A vector space is defined over field F that consists of a set on which two operations (addition and…

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: If the plane F contains the points P(1, 0, 0), Q(3, 2, 0), R(0, 2, 1), L O 1 6 -6 O -2

A:

Q: Show that R³ is spanned by S = {(1,1,1), (1, –1, 0), (0, 1, – 1)}.

A:

Q: (iii) Let X be a set with at least two points. Prove that the discrete space (X, T) and the…

A: Given : X be a set with atleast two points. To prove : Discrete space and indiscrete space ie. (X,τ)…

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: frnd an o-thogonal besis for the {(0,2,1),,0,1), (lo,4.0), (4,2,2){ space spanned by dac set

A: