Let f,g ∈ L∞[a, b] and c ∈ R.a) Show that cf ∈ L∞[a, b].b) Show that f + g ∈ L∞[a, b]. (12) Let f,g € L a, b] and ee R.a) Show that cf € L [a, b).b) Show that +g€L* a, b].

This is a problem of Functional Analysis. To show the closure property of L infinity measure space…

Answered: Let f,g ∈ L∞[a, b] and c ∈ R. a) Show…

Math

Advanced Math

Let f,g ∈ L∞[a, b] and c ∈ R. a) Show that cf ∈ L∞[a, b]. b) Show that f + g ∈ L∞[a, b].

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.5: Graphs Of Functions

Problem 56E

See similar textbooks

Related questions

Q: Consider the set P = {f€C([0, 1], R) | ƒ is nondecreasing} Determine if (P,d2) is separable. Here,…

Q: Suppose A ⊂ R is both closed and bounded, prove that A has the Heine-Borel Property

Q: Define f : Zmn → Zm × Zn by ƒ ([x]mn) = ([x]m, [x]n). Show that f is a function and that f is onto…

A: The function f:ℤmn→ℤm×ℤn is defined by fxmn=xm,xn Show f is a function and f is…

Q: Suppose M and N are C manifolds, U an is C. Show that there exists a neighborhoo that F can be…

A: Let us start solving the problem step by step

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

A: True

Q: Let T be self-adjoint on a finite-dimensional inner product space V. Prove that Vr E V, ||T(x)± i||?…

A: The complete solutions are given below

Q: Let A ϵ Mnxn(F). Under what conditions is det( - A) = det(A)?

A: To Determine: Let A ϵ Mnxn(F). Under what conditions is det( - A) = det(A)? Given: we have a matrix…

Q: prove that an is bounded above.

A: In this question, we prove the sequence an=(1+1n)n is convergent by given porcedure. i.e.…

Q: Use the definition of the dot product a b = |a||b| cos 0 to prove the Cauchy-Schwarz Inequality: |a…

Q: Prove that if r is a vector-valued function that is continuous at c, then || r || is continuous at…

A: Consider the vector r→a=p(a)i+q(a)j+s(a)k. If r is continuous at a=c. So, lima→cr→(t)=r→(c) Then…

Q: Exercise 3 Define f : Rª\{0} by f(r) = |r|,'. Using the e-5-definition, prove that f is continuous.…

A: We are given a function f whose domain is ℝd-{0}. The function f is defined as f=x2-1. Where…

Q: If M is a closed linear subs r space X, and x, be a vect the distance from x, to M, ional f, in X'…

Q: (d) Consider the set [1, √5] C [-1, √10]. Explain f is not continuous by (i) connectedness, (ii)…

Q: Every continuous function from normed space X into normed space Y need not be bounded linear…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let T be the cuclidean topology on R and r' be the finite-closed topology on R. Define f : (R, 7') →…

Q: let X-j m,n; is Z Ž asigma algebra over X ?

Q: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ such that X-U is either finite or X To check…

Q: A function from ZxZ to Z defined by f(m, n) = m-n is an onto function. True False

A: Here we will examine that f is onto or not.

Q: Given any Hermitian space E with › E, if (f(x), x) = 0 for all x € E, th I

Q: Let X be a finite dimensional norm space. Then prove that M = {x € X |||x|| <1} is compact.

Q: Let R be equipped with the standard topology. Define q : R? → R, q((x1, x2) : Jx1, if x1 2 x2 |X2,…

A: Question: The function q:ℝ2→ℝ defined by qx1,x2=x1, if x1≥x2x2, if x1<x2 where ℝ2 and ℝ are…

Q: Show that if a non-zero element in a normed space X, then there is a bounded linear functional f on…

A: We need to show that if a non-zero element in a normed space X, then there is a bounded linear…

Q: Prove that ℝ with lower limit topology is Hausdorff

A: Solution :-

Q: Let T be the co-finite topology on R and d0(x,y)=|x-y|. Prove that the function f:(R,T) to (R,d0)…

A: Given that T be the co-finite topology on ℝ and d0x,y=x-y To prove that the function f:ℝ,T→ℝ,d0…

Q: Let (X, d) be a compact metric space and let A ⊆ C(X; C) = {f : X → C; f is continuous} be an…

A: please see the next step for solution

Q: E be a subset of a normed space X and Y = Span E and a € X. Show that Let a € Ỹ if and only if f(a)…

A: To find: a∈Y¯, and f(a) = 0 whenever f∈Xι and f is 0 everywhere on E.

Q: (d) Consider the set [1, √5] C [−1, √10]. Explain f is not continuous by connectedness, compactness,…

A: Given: The function, f:-1,10→ℝ defined by, fx=x2,x>2x,x≤2 The set 1,5⊂-1,10. To explain: Why the…

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

Q: Prove that Cl(Q) = R in the standard topology on R. %3D

A: The closure of S may equivalently be defined as the union of S and its boundary, and also as the…

Q: Prove A = BH (x)A = (Vx)B without using the equivalence theorem.

Q: The vector space C[a, b] of all continuous real-valued functions on [a, b] is not the span of any…

A: A vector space is a set whose elements, often called vectors, may be added together and multiplied…

Q: Use the Heine-Borel theorem to show that every closed and bounded interval [a, b] with the absolute…

A: Compact Set: A collection A of subsets of space X is said to be cover X or covering of X if the…

Q: Examine the continuity of if 0 a at

A: We need to examine the continuity of the given function f(x) at x=a.

Q: Let f: R → R with the usual topology that sends compact sets to compact sets in R ^ 2 Is the…

A: For any x,y∈R if y=fx there there exists a number x±∈R in neighborhood of x such that y±=fx±.

Q: Let {Xa | a E A} be a collection of Hausdorff topological spaces. Prove that IαEA Xa is Hausdorff…

Q: Show that, with z = Rw, for some bounded w, || z(jw)|| w llw(jw)| ||R||. = sup means that -

Q: Consider the linear space C[0, 1] equipped with the norm ||f || = max |f(t)|. te[0,1] Prove that…

A: Given: The linear space C0,1 equipped with the norm, f=maxt∈0,1ft To prove: There is no inner…

Q: Let R have its usual topology and let X = {a, b, c}. Define f : R → X by x 0 а, f(x) = с, Is X with…

Q: Show that if a contraction map f: from M to M has a fixed point then it's unique.

A: Definition: Let X, d be a metric space. The map T: X→X is said to be a contraction mapping if there…

Q: Exercise 6. Prove that if f [a, b] : interval. → R is continuous, then f([a, b]) is a closed and…

A: 6. Given that f : a,b→ℝ is a continuous function. To find: f a,b is closed and bounded interval.…

Q: Let X, Y be normed linear spaces. On X x Y define || (x,y) |l, = || × |lx + || y lly and + || y ily…

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: |A:k||2||Br:||2) < ||A||F||B||F k=1

A: answer is in next step

Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

Q: A mapping f:C→ C is called bounded, if sup |f(2)| < o0. zEC Show that sin, cos : C →C are unbounded…

Q: Let R be the ring of continuous functions from R to R. Prove that M = {f e R : f(0) = 0} is a…

A: To show that the given subset M of R is a maximal ideal, we first show that M is a ideal of R and…

Q: Exercise 16: Prove the following: Theorem 30: Let a, b ∈ Z where a is not equal to 0. Then Rem(b,…

A: Solution:

Q: Consider E = C( [o,(]) as normed spaces with normS II f | = sup 1fcxl and l fll,=Sifwildx. %3D Show…

Question

Let f,g ∈ L∞[a, b] and c ∈ R.
a) Show that cf ∈ L∞[a, b].
b) Show that f + g ∈ L∞[a, b].

(12) Let f,g € L a, b] and ee R.
a) Show that cf € L [a, b).
b) Show that +g€L* a, b].

Expert Solution

This question has been solved!

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

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps with 1 images

SEE SOLUTION Check out a sample Q&A here