3.Two norms |||| and ||| ||| on avector space are equivalent if thereexists c>o such that// | ||||| x |||≤|x|| ≤ c |||*|||,xe t(a) Show that the norms || || and |||.I are equivalent if and only ifthe identity map of is bicon-tinuousbetweenthe.||-||-topology of|||-topology of *.and the |||(b) Show that the norms |||| and||| ||| are equivalent if and only.if sup{|||* ||| || x || = 1} <∞ andinf{||| x ||| ||x|| = 1} > 0.(c) Prove that the notion of beingequivalent norms is an equiv-alence relation on the set ofnorms on *.

Given: In this question, the given details are, There are two norms are present, · and · on the…

- C 3. Two norms |||| and ||| ||| on a vector space are equivalent if there exists c o such that ||| x ||| ≤|x|| ≤ c ||x, xe t Show that the norms || || and ||| • ||| are equivalent if and only if the identity map of is bicon- tinuous between the (a) ||-||-topology of |||-topology of *. (b) Show that the norms || || and ||| ||| are equivalent if and only if sup{|||* ||| || x || = 1} <∞ and inf{|||*|||||||=1} > 0. (c) Prove that the notion of being equivalent norms is an equiv- alence relation on the set of and the ||| norms on .

- C 3. Two norms |||| and ||| ||| on a vector space are equivalent if there exists c o such that ||| x ||| ≤|x|| ≤ c ||x, xe t Show that the norms || || and ||| • ||| are equivalent if and only if the identity map of is bicon- tinuous between the (a) ||-||-topology of |||-topology of . (b) Show that the norms || || and ||| ||| are equivalent if and only if sup{||| ||| || x || = 1} <∞ and inf{|||*|||||||=1} > 0. (c) Prove that the notion of being equivalent norms is an equiv- alence relation on the set of and the ||| norms on .

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Prove or disprove whether the sets S defined below are vector spaces. (i) S= {P(x)|P(x)= ax + bx?…

Q: 1.9. Consider the space R² with the norm ||- ||p, introduced in Section [1.5 For p=1,2,∞o draw the…

A: The vector space ℝ2 is equipped with the norm .p. The norm .p is defined as xp=x1p+x2p1p for p≥1,…

Q: Bolzano-Weirstrass theorem for ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 -…

Q: 1. Let X = =R2 be a two dimensional real vector space and let A be the matrix Define a mapping T :…

A: ∣∣T∣∣=1Explanation:To find the operator norm ||T|| induced by the ∣∣⋅∣∣1 norm, we need to find the…

Q: Let X, Y be the non-zero normed spaces. Then for any non-zero element x = X, there is a bounded…

A: Let's construct the linear map T:X→Y described as follows.Since x is a non-zero element of X, let's…

Q: = Vi O Let T be the linear operator on V that satisfies T(U(v;)) (1 ≤ i ≤ k) and T(w;) = 0 (1 ≤ i ≤…

A: In this solution, we will consider a linear operator U on a finite-dimensional inner product space V…

Q: (a) Let (V, (,)) be a real inner product space, and let || . || : V → R be the norm derived from the…

Q: 1. Let V be an inner product space. Suppose f,g E V such that f -gg (a) Prove that 1I2 = |lf- gll2 +…

A: To prove the required norm property in an inner product space, under the given conditions

Q: (a) Formulate and prove the theorem on boundedness and continuity of linear operators on normed…

A: (a) Theorem on Boundedness and Continuity of Linear Operators on Normed Spaces: Let X and Y be…

Q: Proposition 18. Norms are conver. Proof. Let || - || be a norm on a vector space V. Then for all x,…

Q: 1. We let P(R) denote the set of all polynomials of degree at most n with real coefficients. Let T =…

Q: Check that the following are vector spaces by proving in full detail that all axioms (one by one) in…

A: 1. a) Let u, v, w ∈Rn ; so u= (u1,u2,u3,......,un) ,v=(v1,v2,v3,........,vn), w=…

Q: Theorem 9.6.1. Let V be a real vector space. A norm ||· || is induced by an inner product if and…

A: We have to prove that a norm · is induced by an inner product if and only if, for all x,y∈V , the…

Q: a normed space, mapping such that ker(T) = {0}). Show that: defines a norm on E. any vector space…

A: Answer given below.

Q: 4. Show that the norm on l0 given by I| (11, .., In...) |lo= sup{|r;| : j= 1,2, ...} can not be…

Q: Let P be a projection of a normed lined

A: Given that P→Projection X →Normed Linear Space To prove: Result 1: P is closed map if and only if…

Q: 3. Let V be a finite dimensional vector space and let (.,.,) be an inner product on V. Suppose that…

Q: (1) Let (X, 7) be a topological space and let A CX. Let TA= {UNA: U ET}. (a) Show that TA is a…

A: Topology: Let τ be a collection of subsets of a nonempty set X. Then τ is called a topology if this…

Q: Let v,w ɛ R" and define A = vw². Prove that ||A||2 result true for the Frobenius norm, i.e. is…

A: Let v,w∈ℝn define A=vwT. v=x1,x2,...,xn and w=y1,y2...,yn. Then,…

Q: Exercise 2. Let X be a topological space. Endow Xx X with the product topology. Consider the map f:…

A: We have to give an example of a topological space X such that the diagonal is not closed in X×X

Q: Rocky Mountain Chocolate Factory (RMCF) founder and president Frank Crail employs 220 people in 361…

A: for net price equivalent rate determine first find the complement of the chain discount rate…

Question

100%

3.
Two norms |||| and ||| ||| on a
vector space are equivalent if there
exists c>o such that
// | ||
||| x |||≤|x|| ≤ c |||*|||,
xe t
(a) Show that the norms || || and |||
.
I are equivalent if and only if
the identity map of is bicon-
tinuous
between
the
.
||-||-topology of
|||-topology of *.
and the |||
(b) Show that the norms |||| and
||| ||| are equivalent if and only
.
if sup{|||* ||| || x || = 1} <∞ and
inf{||| x ||| ||x|| = 1} > 0.
(c) Prove that the notion of being
equivalent norms is an equiv-
alence relation on the set of
norms on *.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Trending now

This is a popular solution!

Step by step

Solved in 5 steps

SEE SOLUTION Check out a sample Q&A here

Follow-up Questions

Read through expert solutions to related follow-up questions below.

Follow-up Question

Hello. Can you hand write the solution. I'm not understanding. It's not really clear. Thank you

Solution

by Bartleby Expert

SEE SOLUTION

Similar questions

2. Let X be a normed space. For each element (x, y) = XX, put || (x, y)||₁ == ||x|| + ||y|| and || (x, y) || = max(||||, ||y||). Show that || ||1 and || are equivalent norms on XOX. . 2 Show that if (a) is a coont seguonoo in I then it is also
Prove that |,| is a valid norm
6. Suppose V is a real inner product space. (a) Show that if u, v € V with ||u|| = ||v||, then u + v is orthogonal to u – v. (b) Provide an example of a real inner product space V and u, v Є V such that u + v is not orthogonal to u - v.
5. Let V be a vector space (note that this is an arbitrary vector space - not necessarily F"). Suppose dim(V) = n. Prove that a set of vectors (v₁,..., Vn} in V is linearly independent if and only if it spans V. • Hint 1: Remember (from sec. 1.6) that there is always an isomorphism A: V → F and isomorphisms "preserve" bases (and so preserves linear independence). Hint 2: Think in terms of pivots.
Prove or disprove whether the sets S defined below are vector spaces. (i) S={P(x)|P(x) = a + bx + cx², with a + b+ c=0} (ii) S={A e M, (R)l det(A) = 1}