Sequence of Integrable Fns Converging to Integrable Fn But Not in L1-Norm

In summary, the conversation discusses an example of a sequence of integrable functions that converges to an integrable function, but not in the L1-norm. The example given is fn(x) = n2xe-nx, where the integral (0,∞) of fn(x) = 1, but the pointwise limit is 0. Another commonly used example is taking f_n to have graphs that are triangles of decreasing width but increasing height and always have one vertex at the origin, resulting in \int |f_n| = \int f_n = 1 \neq \int f = 0.
  • #1
kikkka
1
0
Dear friends can you show me please an example of a sequence of integrable functions fn:R->R converging to an integrable function f but *not* in the L1-norm, i.e. such that
\Int \mid f_n -f\mid is not equal to 0?
Thank u a lot
 
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  • #2


Let fn(x)=n2xe-nx. The integral (0,∞) of fn(x) = 1, but the point wise limit = 0, so we don't have L1 convergence.
 
  • #3


I think the canonical example here is taking your [itex]f_n[/itex] to have graphs that are triangles of decreasing width but increasing height that always have one vertex at the origin, so that the pointwise limit is 0 (give me any [itex]x\in \mathbb R[/itex], and I will make [itex]f_n(x) = 0[/itex] for all [itex]n[/itex] sufficiently large), but [itex]\int |f_n| = \int f_n = 1 \neq \int f = 0[/itex].
 
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FAQ: Sequence of Integrable Fns Converging to Integrable Fn But Not in L1-Norm

What is a sequence of integrable functions?

A sequence of integrable functions is a series of mathematical functions that can be represented by a definite integral. This means that the function has a finite area under its curve on a given interval.

What does it mean for a sequence of integrable functions to converge?

Convergence in this context means that the functions in the sequence approach a single function as the index (or position) in the sequence increases. In other words, the functions become more and more similar as the sequence progresses.

What is the difference between convergence in the L1-norm and convergence in the L2-norm?

Convergence in the L1-norm refers to the convergence of the functions' integrals, while convergence in the L2-norm refers to the convergence of the functions themselves. In other words, L1-convergence is based on the area under the curve, while L2-convergence is based on the values of the functions at each point.

Can a sequence of integrable functions converge in the L1-norm but not in the L2-norm?

Yes, it is possible for a sequence of integrable functions to converge in the L1-norm but not in the L2-norm. This means that the integrals of the functions approach a finite value, but the functions themselves do not converge to a single function.

What is an example of a sequence of integrable functions that converges to an integrable function but not in the L1-norm?

An example of such a sequence is the sequence of functions f_n(x) = n * x^n on the interval [0,1]. Each function in the sequence is integrable on this interval, but the functions do not converge in the L1-norm because their integrals approach infinity as n increases. However, the functions do converge pointwise to the function f(x) = 0 on this interval.

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