Llinear functional on N U 0
Not what you're looking for?
Let H = l^2(N U 0)
(a) Show that if {a_n} is in H, then the power series sum_{n=0}^infty a_n z^n has radius of convergence >= 1.
(b) If |b| < 1 and linear functional L: H-->F (F is either the real or the complex field) is defined by L({a_n}) = sum_{n=0}^infty a_n b^n, find the vector h_0 in H such that L(h) = < h, h_0 > for all h in H.
(c) For a bounded linear functional L: H-->F define the norm of L as follows:
||L|| = sup {|L(h)|: for all h in H such that ||h||<1 }. What is the norm of the linear functional L defined in (b)?
Purchase this Solution
Solution Summary
In this stepwise solution, the calculations are given for finding the vectors and norms of the linear function.
Solution Preview
(a) The fact that H is l^2 means that the sum_0^infty |a_n|^2 converges; therefore |a_n|-->0 as n->0.
Therefore the power series sum_{n=0}^infty a_n z^n ...
Purchase this Solution
Free BrainMass Quizzes
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.