Consider the linear space C[0, 1] equipped with the norm|| ||= max f(t)|.te[0,1)Prove that there is no inner product on C[0, 1] agreed with this norm.

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

Answered: Consider the linear space C[0, 1]…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let V be a finite-dimensional inner product space over C with inner product (, ), and let a, b e V \0. Define T: V + V by T(v) = (v, a) b for all v E V. (i) Show that T is a linear map. (ii) For v E V, find T*(v) in terms of v, a, b. (iii) Prove that if T = T*, then b = Xa for some A E R.
1) Let V, W be F vector spaces. Recall that a skew-symmetric map is a function f: V × V → W such that ƒ(V₁, V2) = −ƒ (V₂, V₁) (a) (b) Prove that an alternating map is always skew-symmetric Prove that if char(F) + 2 then a skew-symmetric map is also alternating
3. Let V = P3, the vector space of polynomials of degree at most 3. Let B {1, t, t², t³} and C = {t³,t,t², 1} be bases for P3. If f(t) = (1+ t)(t² + 4t + 1). Find [f(t)]8 and [f(t)]c
4. Let V = {ao+aj x+a2 x² | ao, a1, a2 E R} be a vector space of polynomial up to degree three, and p(x) = pPo+P1 x+p2 x², q(x) = qo+q1 x+q2 x² E V then show that = po.4o + pP1-q1 + P2-92 defines an inner product on V.
(1) Consider v = (v₁) € R" (v; are the components of v). (la) standard (₂) vector norm ||v|| = √vTv (1b) the maximum norm ||v||max = max; [v₁], where v; are the components of v = (v₁) (1c) the ₁ vector norm ||v||₁=;|v₁| Compute these norms for v = √2
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Let P3(R) be the vector space of all real polynomials of at most degrees 3 with the inner product defined as (f,g) = [ * f(x)g(x)dx. (a) Find an orthonormal basis B for P3 (b) Consider the linear operator T : P3 → P3 defined as T(p) = p". Find the charac- teristic polynomial and the minimal polynomial for T. (c) Find the matrix of T from part (b) with respect to B from part (a).
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For x = {Xn}neN € l¹ (R), set || x || = sup neN Show that or give a counterexample for: (i) (e¹ (R), I-II) is a normed vector space; (ii) (e¹ (R), ||-·||) is a Banach space; (iii) |||| is equivalent to ||-||1. ΣX₂. Xk
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Consider the linear space C[0, 1] equipped with the norm ||f || = max |f(t)|. te[0,1] Prove that there is no inner product on C[0, 1] agreed with this norm.