Define an inner product space for continuous functions on [0, 1] as(S, g) = [f(x)g(x)ax with the usual norm.If f(x) = 2, g(x) =x+1 and h(x) =1-xa) Find the angle e between g and hb) Apply the Gram-Schmidt process to find the orthonormalize set of these functions.

Answered: Image /qna-images/answer/007a38fb-a68b-4ac9-b815-e83640c08d8f.jpg

Answered: Define an inner product space for…

Define an inner product space for continuous functions on [0, 1] as (S, g) = [f(x)g(x)ax with the usual norm. If f(x) = 2, g(x) = x+1 and h(x) =1-x a) Find the angle 6 between g and h

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section: Chapter Questions

Problem 12T

See similar textbooks

Related questions

Q: is continuous at (0,0). 3. Show that X = [0, 1] with the discrete metric is bounded but not totally…

Q: If a subadditive functional p on a normed space X is continuous at 0 and p(0) = 0, show that p is…

A: Given: A sub additive functional p on a normed space X is continuous at 0 and p0=0. To prove: p is…

Q: Show that a uniformly continuous function on (a, b) is bounded.

Q: Consider the vector space V = C([0, 1]) of complex-valued functions that are continuous on [0, 1].…

A: To establish the Cauchy- Schwarz inequality for the functions in C[0,1]= the space of complex valued…

Q: Let ƒ be Riemann integrable on [a, b], and suppose g : [a, b] → R is a function such that g(x) =…

A: Given information f be a Riemann integrable on a,b g:a,b→R is a function such that fx=gx except for…

Q: Let f be a function defined from a topological space (X, T) to atopological space (Y,T,).Let a be a…

A: Let f:X,Tx→Y,Ty be a function and a is open in Tx. To prove that the function is continues at a.…

Q: Let V be the vector space of differentiable functions on 0, 1| with continuous derivative. Which…

Q: ɔ that h is a continuous functi (x2 — 2, х < 6 h(x) 3= 7х + a, х 6

Q: A function F is defined on R² By: f (x, y) (x, y)# (0,0) F (x, y) = %3D (x, y) = (0,0) x²y+2xy² Let…

Q: 3. Show that every y E lị defines a continuous linear functional on l through the inner product,…

Q: determine if the limit exists

A: In this question, we find the limit in two variables by using different paths.

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: Let f:X → Y be function and g: X → fcX ×Y, g(x) = (x, f(x)). Prove that if g is a homeomorphism then…

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

Q: Let f be a continuous function with domain R. Compute the limit (z- t) f(t) lim dt.

Q: 4. Prove that a function which is harmonic everywhere on a plane and is bounded either above or…

Q: (a) Assume that g(x, y) is a function of the form g(x, y) = {" 0 if (x, y) = (0,0). (i) Show that…

Q: Let f: X_nte, Y be X,Y are Tj spaces. show that: Xö locally compact iff Y is so . perfect function,…

A: Given: f:X→ontoY is a perfect function, where X,Y are T2-spaces. To show that X is locally compact…

Q: Derive a finite difference approximation to of(x,y) + f(x,y) using the points f(x, y), ƒ (x+h, y),…

A: Here we use taylor series to find this.

Q: Let f:X Y be function and g:X→fcXXY, g(x) = (x, f(x)). Prove that if g is a homeomorphism then f is…

A: Given that, f:X→Y be function and g:X→f⊂X x Y defined byg(x) = (x,f(x))Given that, 'g' is…

Q: Let CR be the space of all continuous and bounded functions x(t) on R with norm ||x|| = supR |x(t)|.…

A: As per the question we are given a normed space over the field CR and an operator A such that :…

Q: Show that with the subspace topology Uy on Y, the inclusion map : Y → X is continuous.

A: Given that, UY is subspace topology on Y. Suppose (X, Tx) be topological space and Y is subset of X.…

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: If R is a closed region on the xy-plane andf is a continuous function on R, then f has a relative…

A: The given statement is True.

Q: 8. C'(0, 1] is the space of continuous functions f : [0, 1] → R with a continuous derivative f' :…

A: Let, fnn=1∞ be a Cauchy sequence in C10,1than for any ε>0 there exist N∈ℕ such that ∀m,n>N…

Q: Let f : [0 , 1] → R be a continuous function on [0 , 1] such that : f(x)x" dx = 0, (n = 0, 1, 2,…

A: Given that f:0,1→R be a continuous function on 0,1 such that ∫01f(x)xndx=0. The objective is to show…

Q: Let f:X Y be function and g:X →fcX XY, g(x) = (x, f (x)). Prove that if g is a homeomorphism then f…

A: Given : g: X→f⊂X×Y is a homeomorphism. So, by definition of homeomorphism, g is continuous also.…

Q: Let AC C([-1,1]) be defined by A = {f€c'(I-1, 1), f(0) = 0, f'(x)| < 1, V x€ (-1,1)} Prove that A is…

A: The set A is defined as A=f∈C1-1,1, f0=0, f'x≤1, ∀x∈-1,1. To prove that A is relatively compact.…

Q: Let f be a bounded function on [a,b] and suppose that (P)aEN is a family of partitions on [a,b]such…

A: Given: Let f be a bounded function on [a,b] and (Pn)n∈N is a family of partitions on [a,b] such that…

Q: Prove that f(2) = z/(z* + 1) is continuous at all points inside and on the unit circle Iz] = 1…

A: we need to find those points where the denominator ( z^4+1) equals to 0

Q: 5. Determine whether the closed interval [a,b]is homeomorphism to the closed unit…

A: This is a problem from topology.

Q: 2) Let X be normed space and xo + 0 be an element of X. Then using Hahn-Banach Theorem (Normed…

A: We will construct a functional satisfying given data

Q: Let F be defined on R x R to R by F(x, y) = x² – y. Show that F is C' on a neighborhood of (0, 0)…

A: Let F be defined on RXR→R by F(x,y) =x2-y. Now we have to show that F is in class C' on a…

Q: Show that the function f defined by 1, (x, y) = (1, – 1) f (x, y) : .2 x + y (x, y) # (1, – 1) x + y…

Q: 1. (a) Let f: X → R be a subadditive function defined on a topological vector space. Prove that if f…

A: As you have asked more than one different/unreated to each other questions. We shall solve first…

Q: If f(x) and g(x) have a period p, show that h(x)3Da f(x) + b g(x) (a, b constants) has the period p.…

Q: ) Prove that the envelope of the family of f (x, y, a) = 0 member of the family at a characteristic…

Q: (b) Let f be a function defined by 1 if x e Qn [a,b] f(x) = - |-1 if x€ (÷) n [a, b]. Show that fis…

A: we are given that f(x)=1 if x∈Q∩[a,b]-1 if x∈(RQ)∩[a,b] To prove f is not Riemann-integrable on…

Q: Let C[0, 1] have inner product [f(x), g(x)] = ∫ f(x)g(x) dx. Find α so that (x + αx2 ) ⊥ (x + 1)

Q: Suppose that ƒ is a Riemann integrable function on [a, b]. Prove, in two lines only, that if f is…

A: Bounded function

Q: Prove that the limit of xy[(x^2 - y^2)/(x^2 + y^2)] is zero as (x, y) goes to (0,0)

Q: Let f(x) = sin x and g(x) = cos x, and W = span{f,g}. Use the Wronskian to show that f and g are…

Q: Let f : AC X → Y be a continuous function between metric spaces. If A is open then so is f(A).

A: Continuous Function: Let (X, dX) and (Y, dY ) be metric spaces. A function f : X → Y is continuous…

Q: (b) Let a > 0, and let X = C (|-a,a];IR) be the space of continuous real-valued functions defined on…

A: The contraction mapping theorem is very significant in topology. A map T from a metric space to…

Q: A set of real numbers D is called dense if and only if every open interval contains at least one…

Q: Compute the value of f.dr If F= -yi+xj and c is a from (-1,1Y to CI,1) %3D SemICirele

A: Proceed as shown below

Q: Use the inner product 1 (f, 8) = f(x)g(x) dx in the vector space CO[0, 1] of continuous functions on…

Q: Show that the space C(R) of all continuous functions defined on the real line is an…

A: According to the given information, it is required to show that C(R) of all continuous functions…

Q: b) Let f(x) = 1 and g(x) = x be functions in C[0, 1]. Use the inner product defined in Example 5, to…

A: Note: Since there is no proper instruction is given and the first part is incomplete, we have solved…

Question

Define an inner product space for continuous functions on [0, 1] as
(S, g) = [f(x)g(x)ax with the usual norm.
If f(x) = 2, g(x) =x+1 and h(x) =1-x
a) Find the angle e between g and h
b) Apply the Gram-Schmidt process to find the orthonormalize set of these functions.

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

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Numerical Differentiation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Verify divergence theorem for f=x2i+zi+yzk taken over the cube bounded by x=0,x=1,y=0,y=1,z=0 and z=1
Suppose that the second order partial derivatives, fx,y and fy,x, are both continuous on an open set V in R2. Use Fubini’s theorem to prove that fx,y = fy,x in V . Hint: if fx,y(a) − fy,x(a) > 0, there is a rectangle R containing a on which fx,y − fy,x > 0.
Consider the operator L^ = k d/dx, where k is a scalar, operating in a function space on the interval[a, b] with inner product defined by <f/g>= ∫ba f*(x)g(x)dx a) Show that Lˆ is a linear operator, i.e., that is satisfies the necessary requirement for linearoperators.b) Find Lˆ† and using this find the conditions on k that make Lˆ self-adjoint.c) Find two sets of boundary conditions under which Lˆ is Hermitian. (if unclear see image)
Let f be a bounded function on [a,b]. Prove that f is integrable on [a,b] if and only if there is a sequence of partitions {Pn} of the interval [a,b] such that limn→∞ (U (f,Pn)) - L(f,Pn) = 0
Prove that if f and g are Riemann integrable on [a; b], then f · g andf + g are Riemann integrable on [a; b].
Show that if f is Riemann integrable on [a,b] and f(x) ≥ 0 for all x ∈ [a,b],then
Use a Wronskian to determine whether the set of functions { 1, cos x, sin x } on (-inf, inf) is linearly independent or linearly dependent.
Consider the subspace W of D, given by W = span(sin x, cos x). (a) Show that the differential operator D maps W into itself. (b) Find the matrix of D with respect to B = {sin x, cos x}. (c) Compute the derivative of f(x) = 3 sin x - 5 cos x indirectly and verify that it agrees with f'(x) as computed directly.
find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = tan x, a = π
Suppose that w and r are continuous functions on (−∞, ∞), W (x) is an invertible antiderivative of w(x), and R(x) is an antiderivative of r(x). Circle all of the statements that must be true.
Show that the Dirichlet function f defined on [0 , 1] by f(x) = {1 if x is rational0 if x is irrationalis not Riemann integrable on [0 , 1]