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Functional Analysis Homework Help | Functional Analysis Assignment Help
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Functional Analysis Topics & Concepts Covered
| Topics | Concepts |
|---|---|
| Normed Spaces | Norms & Metric Spaces |
| Banach Spaces | Completeness |
| Hilbert Spaces | Inner Product Spaces |
| Linear Operators | Adjoint Operators |
| Bounded & Unbounded Operators | Spectrum of Operators |
| Spectral Theory | Riesz Representation Theorem |
| Dual Spaces | Hahn-Banach Theorem |
| Compact Operators | Compact Sets |
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Recently Asked Functional Analysis Questions
Expert help when you need it- Q1:Westside Energy charges its electric customers a base rate of $6.00 per month, plus 15€ per kilowatt-hour (kWh) for the first 300 kWh used and 2€ per kWh for all usage over 300 kWh. Suppose a customer uses x kWh of electricity in one month. (a) Express the monthly cost E as a piecewise defined function of x. (Assume E is measured in dollars.) E(x) = Need Help? (b) Graph the function E for 0 sxs 600. E 60 50 L 40 30 20 10 E 60p 50 40 30 20 10 if 0 sxs 300 Read It if 300 < x 100 200 X 300 400 500 600 100 200 300 400 500 600 Watch It E 100 80 60 40 20 X 100 200 300 400 500 600 E 60 50 40 V 30 20 10 3 100 200 300 400 500 600See Answer
- Q2:-2 QUESTIONS : T₁ -1 €9 6 5 0 1 ५ 1.Based on the graph, identify f(1) = 2. Based on the graph, identify f'(1)= 3 Find T1 tangent formula and result 3 4See Answer
- Q3:-2 QUESTIONS: T₁ -1 Cg 6 5 0 4 1.Based on the graph, identify f(1) = 2. Based on the graph, identify f'(1)= 3 Find T1 tangent formula and result 3See Answer
- Q4:Q1 (9 points) Given a constant c> 0, describe the geometry of |||| = c for x = R², p = 1, 2, 00. For each p, find the solution to min ||||p, subject to 3x₁ + 4x₂ = 1. Explain how to find the solution geometrically for each p. Note: The second part of this question serves to explain why we want to solve an 1₁-norm minimization problem if we want to get a 'sparse' solution (i.e., some elements of the solution are zeros) in some applications.See Answer
- Q5:Q2 (3 points) Let x ER". Show that ||||2 ≤||||1 ≤ √√||||2. Hint: For the 2nd inequality, think about which tool you have learned can be used to show it.See Answer
- Q6:Q3 (4 points) Let A € Rmx Show that||A||₁ ≤||A||2 ≤√|| A||1. Hint: Use the definitions of the related matrix norms and Question 2.See Answer
- Q7:Q4 (4 points) Use the definition of ||- ||2 (not the formula given in the lecture notes) to show that ||A||2 ≤ ||A||f. Hint: Partition A into rows.See Answer
- Q8:Q5 (5 points) (5 pts) Let A € Rmxn and BE Rnxk. First show that ||AB||F ≤ || A||2|| B|| (hint: partition B into columns) and then use this inequality to drive ||A||F√√|| A||2.See Answer
- Q9:Q6 (3 points) Suppose || || is a vector norm. Is || . || defined by ||AT||μ ||A|| := min 40 a counterexample. Hint: Check if || || satisfies the three properties. a matrix norm? Either prove it or giveSee Answer
- Q10:Q8 (4 points) . Use the definitions of ||- ||2 and ||- ||F to show that ||uv||2₂= ||uv||F = ||u||2||v||2.See Answer
- Q11:Q9 (5 points) Let A € Rxn be nonsingular. Write (A) = ||A||·||A¯¹||. (a) Show that K(I) = 1 for any induced matrix norm || - ||. (b) Show that K(A) > 1 for any consistent matrix norm · II. Note: K(A) is called the condition number of A and will be introduced later in this course.See Answer
- Q12:Question 1. [6, 2, 2 marks] (a) Prove that if (,) and (,)2 are two inner products on a real inner product space H with associated norms ||· ||₁ and ||·||2 satisfying . ||f||1 = ||f||2 for all fe H, then (f,g)1 = (f,g)2 for all f, g € H. (HINT: First prove the Polarization identity (f,g) = (f + g||² + ||ƒ − g||²)). (b) Give a geometrical interpretation of (a). (c) For complex inner product space, prove that 4Re(f, g) = (||ƒ+g||²+||ƒ-g||²).See Answer
- Q13:Question 2. [8 marks] Consider the function space (C[0, 1],|- ||p) where ([\ƒ],)³ := f ' \f(x)\Pºdx. Prove that for p = 2, the p-norm || - |, cannot be generated by any inner product. (HINT: Use parallelogram property characterization of inner product norm; use f(x) = x and g(x) = 1- x as counter-examples.)See Answer
- Q14:2. CAUCHY SCHWARTZ INEQUALITY For Cauchy Scwhartz inequality read p19 of 0 Inner Product Fin Dim Question 3.[4, 2 marks] Consider the normed spaces (RN, ||- |1) and (RN, ||- ||2). Prove that: (a) x1 < Mx||2 for all x € RN for some constant M > 0. (HINT: Use the Cauchy-Schwartz inequality in the inner-product space (RN, ||-||2).) (b) Give a geometrical interpretation of (a). 1See Answer
- Q15:Question 4.[2, 1, 2, 1 marks] Prove that (₁, 1) (l2, ||- ||2) as follows: (i) Let B., (0; 1) denote the open unit ball {x € l₁ | ||×||1₁ < 1}. Prove that it suffices to show that B|||₁ (0; 1) ~ (l2, ||· ||2₂). (ii) Suppose x := (₁, 2, n,) belongs to B₁.₁(0; 1). Prove that each com- ponent x₁ <1 for all i € N. (iii) Prove that x2 is finite by showing that ||x||2 < |x||1₁. (iv) Give a geometric interpretation.See Answer
- Q16:Question 5.[3, 5 marks] Let X₁ := {fe C[0, 1]f(0) = 0}. Prove that: (i) XA is a subspace of (C[0, 1], (ii) XA is incomplete.See Answer
- Q17:1. EXTENDING THE NORM Question 1. [3, 5, 6 marks] (a) Let 1 < p < 0. In third year Real Analysis course it was shown that (R", || ||p is a normed space. Using this fact prove that the sequence space (lp, || ||p) is a normed space. That is, for any pair of vectors x := (n)-1 and y := (n)-1 with -1 |xn|P < 00, (i) xp = 0 ⇒ x = 0. (ii) ||Ax|p = |A|||x|p. (iii) x + yp < |x|p+y|p. (Prove each of the norm properties.)See Answer
- Q18:Question 2. [2, 5, 4; 3 marks] (a) Consider the normed spaces (p,|| . ||p) where 1 < p <00. Prove that the canonical basis B := {en\n\ basis as follows: 1,...,} is a Schauder (i) Prove that the set 3 is linearly independent. (ii) Prove that ∞ X = - Σxiei i=1 = (iii) Prove that for loo, oo) that there is no Schauder basis. (HINT: Use a counterexample.) (b) Consider the normed spaces (co, ) of all sequences with limit equal 0. Prove that the canonical basis 3:= {enn = 1,...,} is a Schauder basis for co.See Answer
- Q19: 2. Assume that P = (((p1^-P2) AP3) ⇒ P2). Write down the truth table for P. How many different interpretations make P true. Give the list of interpretations I under which P false. 3. A formula P is called satisfiable if there exists an interpretation I such that I(P) = 1. Which of the following formulas are satisfiable? Check using truth tables. \text { (a) }\left(\neg p_{1} \wedge p_{2}\right) \text { (c) }\left(p_{1} \Leftrightarrow \neg p_{1}\right) \text { (d) }\left(p_{1} \wedge\left(\neg p_{2} \vee \neg p_{1}\right)\right)See Answer
Popular Subjects for Functional Analysis
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