It can be shown that the sequence {fk} infinity, k = 1, converges to 0
Pointwise on [0, +oo) but convergence is not uniform on [0, +oo).
a. Show that convergence is not uniform on [0, 1] either. Thus the sequence does not converge in the normed vectorspace (C ([0, 1]) , II·II infinity).
b. Does the sequence converge to 0 in the normed vector space (C ([0, 1]) , II·II 2 (subscript 2) )?. Even though you can compute the relevant integrals manually, it may be easier to make use of a computer algebrasystem.
Solution Preview
Problem:
Let , k = 1, 2, 3,........
It can be shown that the sequence {fk } infinity , k=1, converges to 0
pointwise on [0, +oo) but convergence is not uniform on [0, +oo).
a) Show that convergence is not uniform on [0, 1] either. Thus the sequence does not converge in the normed vector space (C ([0, 1]) , II·II infinity).
b) Does the sequence converge to 0 in the normed vector space (C ([0, 1]) , II·II 2 (subscript 2) )?
Solution:
a) In order for the convergence to be uniform on [0, 1], one needs to prove that
...
Solution Summary
A pointwise convergences of uniforms are examined. The sequence converge to 0 in the normed vector spaces are determined.
... With metric defined by equation (0.1) it means simultaneous (uniform) convergence for each argument, f_n (t ... Next we show that for any 〖f∈M〗_K in its every ...
... The yellow cell shows the polynomial value when ... In contrast the cubic spline results show high degree ... Furthermore, experience has shown that using splines of ...
... Explain the accounting equation and prepare a table showing the equation and show a list ... clarifies many areas where the company has showed strength or ...
... Subsequent traverses showed that the bifurcation was ... image of downstream traverse showing unstable interface. ... over topography, our observations show that such ...
... A closer analysis (results not shown in Table 3) of ... Studies (eg, Welch and Leege, 1991) show that American ... These analyses showed that Catholic opinion at the ...
... exploratory criteria that the IIG showed constant radius ... 466 INTERPERSONAL MEASURES SHOW CIRCUMPLEX STRUCTURE 467. ... structure, once they are shown effective, is ...
... recognize that the common name of this distribution is Uniform. ... expansion of (1 − a)−b to show that ∞ i ... You may assume convergence for the conditions on a ...
... Our results showed that avatars) and auditory cues (accompanied ... of varying avatar densities, we show the two ... of this work for work topology showing the ports ...
... Norwalk Agreement is the convergence of Generally Accepted ... standards ensuring that statements are correct and uniform. ... These shows the available assets can be ...
