- #1
ScroogeMcDuck
- 2
- 0
Sorry for the rather vague title!
Given:
Problem:
I need to create a sequence [itex]w_n[/itex] in A, for which [itex]w_n \rightarrow 0[/itex] and [itex]Tw_n \rightarrow y[/itex] in B.
So for the sequence [itex]y_n[/itex] of limits of [itex]Tx_i^n[/itex] we know:
[itex]\forall \epsilon>0 \, \exists m \in \mathbb{N}[/itex] such that [itex]\forall n ≥ m: ||y_n - y||<\epsilon[/itex]
And for a fixed n, we know:
[itex]\forall \epsilon>0 \, \exists m_1=m_1(n) \in \mathbb{N}[/itex] such that [itex]\forall n ≥ m_1: ||x_i^n||<\epsilon[/itex]
[itex]\forall \epsilon>0 \, \exists m_2=m_2(n) \in \mathbb{N}[/itex] such that [itex]\forall n ≥ m_2: ||Tx_i^n - y_n||<\epsilon[/itex].
Furthermore T is not necessarily continuous (it would be trivial if it were).
I tried using the sequence [itex]w_n = x_n^n[/itex]. Proving that [itex]Tw_n \rightarrow y[/itex] then required me to prove that [itex]\forall \epsilon>0 \, \exists n' \in \mathbb{N} : \forall n≥n': ||Tx_n^n - y_n||< \epsilon/2[/itex] (so that I could use the triangle inequality afterwards), but I couldn't manage this since y_n is not fixed.
I also tried using [itex]w_n = x^n_{m_3(n)}[/itex] where [itex]m_3(n) = max\{m_1(n),m_2(n)\}[/itex]. The required convergence did work out, but then I realized that m_1(n) and m_2(n) depend on [itex]\epsilon[/itex] as well as n, so my sequence depends on [itex]\epsilon[/itex] which is of course not as it should.
Any suggestions, hints, ideas would be appreciated!
Homework Statement
Given:
- Two Banach spaces [itex]A[/itex] and [itex]B[/itex], and a linear map [itex]T: A\rightarrow B[/itex]
- The sequences [itex](x^n_i)[/itex] in A. For each fixed n, [itex](x^n_i) \rightarrow 0[/itex] for [itex]i \rightarrow \infty[/itex].
- The sequences [itex](Tx^n_i)[/itex] in B. For each fixed n, [itex](Tx^n_i) \rightarrow y_n[/itex] for [itex]i \rightarrow \infty[/itex].
- The sequence [itex](y_n)[/itex] in B, with [itex]y_n \rightarrow y[/itex] for [itex]n \rightarrow \infty[/itex].
Problem:
I need to create a sequence [itex]w_n[/itex] in A, for which [itex]w_n \rightarrow 0[/itex] and [itex]Tw_n \rightarrow y[/itex] in B.
Homework Equations
So for the sequence [itex]y_n[/itex] of limits of [itex]Tx_i^n[/itex] we know:
[itex]\forall \epsilon>0 \, \exists m \in \mathbb{N}[/itex] such that [itex]\forall n ≥ m: ||y_n - y||<\epsilon[/itex]
And for a fixed n, we know:
[itex]\forall \epsilon>0 \, \exists m_1=m_1(n) \in \mathbb{N}[/itex] such that [itex]\forall n ≥ m_1: ||x_i^n||<\epsilon[/itex]
[itex]\forall \epsilon>0 \, \exists m_2=m_2(n) \in \mathbb{N}[/itex] such that [itex]\forall n ≥ m_2: ||Tx_i^n - y_n||<\epsilon[/itex].
Furthermore T is not necessarily continuous (it would be trivial if it were).
The Attempt at a Solution
I tried using the sequence [itex]w_n = x_n^n[/itex]. Proving that [itex]Tw_n \rightarrow y[/itex] then required me to prove that [itex]\forall \epsilon>0 \, \exists n' \in \mathbb{N} : \forall n≥n': ||Tx_n^n - y_n||< \epsilon/2[/itex] (so that I could use the triangle inequality afterwards), but I couldn't manage this since y_n is not fixed.
I also tried using [itex]w_n = x^n_{m_3(n)}[/itex] where [itex]m_3(n) = max\{m_1(n),m_2(n)\}[/itex]. The required convergence did work out, but then I realized that m_1(n) and m_2(n) depend on [itex]\epsilon[/itex] as well as n, so my sequence depends on [itex]\epsilon[/itex] which is of course not as it should.
Any suggestions, hints, ideas would be appreciated!