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L'(0,1). Problem 14. Show that the inner product in a Hilbert space is continuous; that is,...
(4) Let C[0,1] be the inner produce space of all real-valued, continuous functions on the interval (0,1) with inner product.g) = Sopr)(x) dr. Determine the projection of the vector {m} onto the space spanned by the orthonormal system of vectors given below. {1, 73(2x - 1)}
Problem 5. (10 points total) For a linear operator l' on a complex inner product space define (a) (2 points) Show that T+ and T- are self-adjoint and T=T+ +iT-. (b) (3 points) ow that the representation in part (a) is unique. (c) (5 points) Show that T is normal if and only if T+T TT
Please attempt both questions
3. Use the continuous function on the interval (0,1) inner product to find the projection of f(x) onto g(x). (Feel free to use an integral calculator. I use wolfram alpha. Just make sure to type the problem in carefully). (a) f(x) = -22 - 1, g(x) = -2 (b) f(x) = 2r?, g(x) = 2+1 (e) f(x)=-1-1, g(x) = x2 +3 4. Consider 3-space with the dot product. Your subspace S will be the plane z...
2. Consider the vector space C([0, 1]) consisting of all continuous functions f: [0,1]-R with the weighted inner product, (f.g)-f(x) g(x) x dr. (a) Let Po(z) = 1, Pi(z) = x-2, and P2(x) = x2-6r + 흡 Show that {Po, pi,r) are orthogonal with respect to this inner product b) Use Gram-Schmidt on f(x)3 to find a polynomial pa(r) which is orthogonal to each of po P1 P2 You may use your favorite web site or software to calculate the...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
Both part of the question is True or False. Thank you
Problem 1. (ref. Example 3 in the slide) Let X = Y = C[0, 1] (with the norm || ||C[0,1] = sup |u(x)]). For any u € C[0, 1], define T€[0,1] v(t) = u(s)ds. We denote by T the mapping from u to v with D(T) = C[0, 1], i.e., v(t) = Tu(t). Then, the following conditions are true or not? Example 3. We denote by the set of...
Where
Let n(t) be a fixed strictly positive continuous function on (a, b). define H, = L([a,b], 7) to be the space of all measurable functions f on (a, b) such that \n(t)dt <0. Define the inner product on H, by (5,9)n = [ f(0)9€)n(t)dt (a) Show that H, is a Hilbert space, and that the mapping U:f →nif gives a unitary correspondence between H, and the usual space L-([a, b]). We were unable to transcribe this image
5. (15') Define the inner-product on C([-1,1]), the space of all continuous functions on the interval [-1, 1), by (5(2), gla) - s(z)g(z) dr. (a) Use Gram-Schmidt algorithm to convert the set (1,1 + 1,(1 + x)2} to an orthogonal set. (b) Is the set you found in Part (a) still orthogonal if the interval of integral in the definition of inner-product is changed to [0, 1]? Explain your answer.
5. (15') Define the inner-product on C([-1,1]), the space of all continuous functions on the interval [-1,1], by (f(a), g(x) = $ $(a)g(x) dr. (a) Use Gram-Schmidt algorithm to convert the set {1,1 + ,(1+x)?} to an orthogonal set. (b) Is the set you found in Part (a) still orthogonal if the interval of integral in the definition of inner-product is changed to [0, 1]? Explain your an answer.
1). Let V be an n-dimensional inner product space, let L be a linear transformation L : V + V. a) Define for inner product space V the phrase "L:V - V" is an orthogonal transforma- tion". b) Define "orthogonal matrix" b) If v1, ..., Vn is an orthonormal basis for V define the matrix of L relative to this basis and prove that it is an orthogonal matrix A.