a Let R = b beZ}and let o: R→ Z be a mapping defined by : b |= a-b. b a

Q: 8.0. If x-(x,, X, ...,%)«R', define [xL by Prove that xJkL is a norm on R'. N I|} dns - x]

A: We will solve the first question highlighted in yellow. If you want the second question to be…

Q: Evaluate | Vtan a sec" zdx. O a. 2 z tani z +tani +C 2 O b. 2 tani æ + tan +C 2 Ос. 2 O c. 2 2 t…

A: To evaluate the integral,  ∫tanxsec4xdx Rewrite the term sec4x=sec2x·sec2x…

Q: be a topology on Y. Let f be a mapping from X into Y defined by f(a)=f(b) = 3, f(c = 2 and f(d) = 1.…

A: Continuity of functions

Q: Let (R, d) be usual metric space, then Ris ot separable True False

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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…

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Q: Every continuous function from normed space X into normed space Y is bounded linear operator True…

A: True

Q: 12. Show that the set of all continuous function on the interval [-1,1] is a vector space under…

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Q: Let S:x→Y be bounded lineay a operator, Then prove that ker S is closed.

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Q: Find the orthogonal projection of y onto Span{u1,u2}. Where, y=(1,1,0,0)*, u1=(1,0,1,0)t…

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Q: 2. If X,Y are Banach spaces on XOY define |(x, y) = |, +|yly . Show that this gives a norm making X…

A: Given: Dual of c0  is isometric to l1 State the principle of uniform boundedness.

Q: if x=u(1-v),y=(uv) then verify that a(u,v) a(x,y)-1 a(x,y)'a(u,v)

A: Given: x=u1-v,  y=uv To verify ∂u,v∂x,y×∂x,y∂u,v=1

Q: Let X = {a, b, c, d} and TX = {Ø, X, (b), {b,c}, {a,b,d}} be a topology on X. Let Y = {1, 2, 3, 4}…

A: Continuous function- Let X,IX and Y,IY be two topologies. Then a function f:X→Y is said to be…

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

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Q: с) Let A :1, →1, be defined by Ах, х, ...) %3D (0, 0, х2, Хд, .). Prove that A is self-adjoint,…

A: Please find the answer in next step

Q: 8.F. Ifx= (1, I, --- ,)€R", define x, by e that z- l, is a norm on R'. Prove

A: Given that x=x1,x2,..,xn and x1=x1+x2+...+xn. The objective is to find the whether x1=x1+x2+...+xn…

Q: This is for Calc 4, told me to post under advanced math though, please help me solve

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

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Q: Let X be vector space of all ordered pairs. Show that ||x || = |x1| + |x2] defined norm on X %3D

A: It is required to show that: x=x1+x2 defined norm on X, where X be vector space of all ordered…

Q: 5. If r = (r, (x, y, z) and ro = (xo, Yo, 20), describe precisely the set of all points (r, y, z)…

A: We plug the given values in the given equation and simplify

Q: Verify the Cauchy-Schwarz Inequality for u = (1, −1, 3) and v = (2, 0, −1).

A: u = (1, −1, 3) and v = (2, 0, −1) To verify the Cauchy-Schwartz Inequality for the given set of…

Q: Let V, W, Z be vector spaces and let T:V-->W and U:W-->Z be linear transformations. Show that if T…

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Q: There is a vector field F(x, y, z) = (M (x, y, z), N (x, y, z), R (x, y, z)) , such that divF (x, y,…

A: We need to determine the validity of the statement: divF=div(curlF) for a vector field of the form:…

Q: Let R = =3a ala,be Z}and let o: R→Z be a mapping defined by: b = a-b. a

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Q: If T is a one one onto linear transformation (isomorphism) from a vector space U (F) into a vector…

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Q: Justify that (•, ·) defines an inner product on R?. the function (', ·) defined on R? as follows: U1…

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

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Q: = 36 ala,beZ}and let ø : R → Z be a mapping defined by : [a b -6. b

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

Q: Let V(F) be an inner product space and a, Be V. Show that a= B iff ( α, γ)- (β,)V γε V.

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Q: Let V be a real inner product space, and let u, v, w E V. If (u, v) = 1 and (v, w) = 3, what is (3u…

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Q: is an Let V(F) be an inner product space. If 0 + o e V then prove that orthonormal set.

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Q: Let A = Z. Let R be defined on A x A where a Ry if 4 (x- y). Prove that R is transitive.

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Q: Let X = {a, b, c, d} and TX = {Ø, X, {b}, {b,c}, {a,b,d}} be a topology on X. Let Y = {1, 2, 3, 4}…

A: Given topological spaces

Q: 28. Bounded by the planes z = x, y = x, x + y = 2, and z

A: Solution:-

Q: Show that there does not exist a conformal map from C onto the region {z : [2| > 1}.

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Q: If X is a finite dimension that an inner product o Yjk = (e, ex). Can we cho fashion? %3D

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Q: Utu = c²uxx , u(0, t) = 0, u(1, t) = 0, u(x,0) = 0 U¿(x,0) = x² – 2x, 00 t> 0 t > 0 0sx<1.

A: Given P.D.E

Q: Q If X R, define the function ||- ||: R →R*u {o} by ||-|| = |x| + 1. Is (R, ||-||) normed space.…

A: Solution: We will use the following definition: The function .:ℝ→ℝ is a norm if it satisfies the…

Q: Let S = {u, v, w} be an orthonormal subset of an inner product space V. What is ||u + v + w||2?…

A: Solution by using the definition of norm in inner product space and use the distributive property as…

Q: Every continuous function from normed space X into normed space Y is bounded linear operator True…

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Q: If them every normed lineers spare X is finite demisional Linear transformation. X is bounded a

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Q: 9) Suppose u, v € V, where V is an inner product space, and ||u|| = ||v|| = 1 and (u, v) = 1. Find u…

A: I have used the property of norm,  IIxII=0 off x=0

Q: Prove using Venn diagram that An (B U C) = (A n B) U (A n C)

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Q: Once you get away from vector spaces of finite dimension, it is no longer true that every linear map…

A: As per bartleby guidelines for more than one questions asked only first should be answered . Please…

Q: If G (x, y, z) = curlF (x, y, z), where F (x, y, z) = (y – 2, z – x, x – y), then х, х — у), then ||…

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Q: The set R2 is a vector space over R. How Prove it ?

A: We need to check each and every axiom of a vector space to know that it is in fact a vector…

Q: There is a vector field F (x, y, z) such that curlF (x, y, z) = (divF (x, y, z) , (divF (x, y, z))²,…

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Q: Verify that (u.u2) is an orthogonal set, and then find the orthogonal projection of y onto Span (u,…

A: For two vectors to be orthogonal, the dot product should be zero.

Q: Prove that dual space of R^n is R^n( its Functional analysis topic)

A: to prove dual space of ℝn is ℝn. proof let the dual space of ℝn be denoted as ℝn*=M we will show…

Q: Let V be an inner product space with dimV = 2, and suppose that L= {u+v, u+2v} is a linearly…

A: Let V be an inner product space with dimV=2 and L=u+v,u+2v is a linearly independent subset of V.…

Q: Q#1: Discuss Application of inner product spaces with examples.

A: Important Application of Inner product spaces is to find Least squares approximations for a…