Re: Peano curve is wrong

In article <db64c4a9-dcab-4509-9b75-ad93c1ef2bd6@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
qiuzhihong <qiuzhihong@xxxxxxxxx> wrote:
On Apr 6, 3:37=A0am, magi...@xxxxxxxxxxxxxxxxx (Arturo Magidin) wrote:
In article <5471125f-ff2e-43bf-a410-d2b2423bb...@xxxxxxxxxxxxxxxxxxxxxxxxx=
com>,

qiuzhihong =A0<qiuzhih...@xxxxxxxxx> wrote:

This is mostly pointless, seem you seem unwilling (or incapable) of
stating the FULL set of hypotheses being used in the argument. You
seem to believe there is a gap, but that apparent gap only seems to be
there because instead of invoking the correct hypothesis of the
argument, you attempt to invoke a fact that you know to be incorrect
(as does everyone else). It's like arguing that the proof that 1+1=3D2
is incorrect because the argument relies on claiming that a+a is
always equal to 2, which is not true. (Indeed, "a+a=3D2 for all a" is
false, but that is not what is used to prove that 1+1=3D2).

So how about stating ALL the hypothesis, instead of your illusory
ones?

What you said is meaningless. Peano curve is seem as an example of 1-1
mapping between [0, 1] and R^2.

No, it's not, as many people have stated.

If somebody say that in his dictionary 'curve' is equal to the term
'curve or plane' in others understanding, therefore his 'curve' can be
a plane. Is it =A0interesting?

And if my grandmother had wheels, then she would be a bicycle,
something which is also irrelevant to the question at hand.

So: what is a plane curve? In the context of the Peano curve, a plane
curve is defined to be a CONTINUOUS function f:[0,1]-->R^2. Any
continuous function from [0,1] to R^2 is a plane curve.

Now, if you have a sequence {f_n:A->B} of function, and B is a subset
of R^n, you can ask about the "pointwise limit" of the sequence. This
is defined to be the function

g:A->B defined by g(a) =3D lim_{n-->oo} f_n(a).

The domain of g is the set of all a in A for which the limit
exists.

Now, it is NOT the case that for ANY sequence {f_n:[0,1]-->R^2} the
pointwise limit of the sequence is a function whose domain is all of
[0,1]. It is also NOT the case that if each f_n is continuous, then
the pointwise limit is continuous. You are correct in saying that.

However, there are SUFFICIENT conditions to ensure that the pointwise
limit of the f_n IS defined in all of [0,1], and there are also
SUFFICIENT to ensure that if each f_n is continuous, then the
pointwise limit is also continuous.

In particular, there is the notion of "uniform convergence": a
sequence of functions {f_n:A->R^n} is said to CONVERGE UNIFORMLY to a
function f(x) if and only if for every e>0, there exists N>0 such
that FOR ALL a in A and all n>N, |f_n(a)-f(a)|<e.

Compare this with the notion of "pointwise convergence": the sequence
f_n as above is said to CONVERGE POINTWISE to f(x) if and only if for
every a in A, for every e>0, there exists N>0 such that for all n>N,
|f_n(a)-f(a)| < e. It is similar to the difference between being
"continuous" and being "uniformly continuous".

The example of the sequence {f_n:[0,1]->R} given by f_n(x)=3Dx^n is a
sequence that converges pointwise to g(x)=3D0 if 0<=3Dx<1, and g(1)=3D1, b=
ut
it does NOT converge uniformly to this function.

There is a theorem that states:

=A0 UNIFORM CONVERGENCE THEOREM: If {f_n} is a sequence of CONTINUOUS
=A0 functions that converges UNIFORMLY to a function f(x), then f(x) is
=A0 also continuous.

In particular, in the case of the Peano sequence of functions, it is
shown that the sequence has a pointwise limit at every point of [0,1],
and hence that the sequence converges pointwise to a given function
F:[0,1]->[0,1]x[0,1]. It is then ALSO shown that the sequence in fact
converges UNIFORMLY to F. Thus, by the Uniform Convergence Theorem,
the function F must be a CONTINUOUS function from [0,1] to
[0,1]x[0,1]. So F is a curve, not just because it is a limit of
curves, but rather because it is a UNIFORMLY CONVERGENT limit of
curves. This ->is<- established, but is exactly the important point
that you keep skipping over.

Once you know that F is a curve, as a consequence of the Uniform
Convergence Theorem, you establish separately that F is surjective
(it is NOT 1-to-1), and you are done.

If you bother to consider ALL the parts of the argument, instead of
substituting your own gaps into it, at least.


It's you that stating ALL the hypothesis.

Indeed I am.

Do you know what is deduction?

Apparently far better than you have any hope to.

Even if it may has an exception of
Peano's deduction, and he hasn't mention the exception, his deduction
is not a strict one.

Since you do not seem to be able to even articulate what this
conclusion is supposed to be, forgive me if I dismiss your childish
tantrums as little more than silliness on your part.

See these two deductions:

1/ Every curve and [0, 1] have a 1-1 mapping;

This is nonsense as written.

2/ The limit of 'Peano's curve sequence' filling the whole plane;

Nobody claims this.

3/ There is a curve filling the whole plane.

Nobody claims this.


Do you think 1,2 -> 3 is a strict deduction?

I think (1) is nonsense as written, (2) is a false assumption that
nobody is claiming, and you wouldn't recognize a deduction if it bit
you in the ass, hard.

1/ Every number sequence and the natural number set has a 1-1 mapping;

Nonsense as written.

2/ There is a sequence of number sequence, the limit of the 'sequence
of number sequence' filling [0, 1];

Nonsense as written.

3/ There is a 1-1 mapping between natural number set and [0, 1].

True but a non sequitur from your nonsensical first two statements.

Do you know what is fault of deduction?

Do you?

This is fault of deduction.
Too simple, sometimes naive.

It is apparent that you do not know what you are talking about and can
hardly string together a coherent sentence. Perhaps you might want to
start there, instead of exhibiting your simplicity and naivete.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

Relevant Pages

  • Re: Peano curve is wrong
    ... 'curve or plane' in others understanding, ... you can ask about the "pointwise limit" of the sequence. ... UNIFORM CONVERGENCE THEOREM: If is a sequence of CONTINUOUS ...
    (sci.math)
  • Re: Peano curve is wrong
    ... 'curve or plane' in others understanding, ... you can ask about the "pointwise limit" of the sequence. ... UNIFORM CONVERGENCE THEOREM: If is a sequence of CONTINUOUS ...
    (sci.math)
  • Re: Peano curve is wrong
    ... curve" of the sequence P is filling the unit square. ... even if the "limit curve" of Peano curves covers the ... One needs to show that the pointwise limit is continuous. ...
    (sci.math)
  • Re: Peano curve is wrong
    ... curve" of the sequence P is filling the unit square. ... even if the "limit curve" of Peano curves covers the ... One needs to show that the pointwise limit is continuous. ...
    (sci.math)
  • Re: L_p spaces question
    ... Let be a sequence of disjoin measurable sets and for each n let ... You need some extra hypothesis for the last statement. ... convergent sequence in a banach space is convergent (that is f exists ... of course one has to show that the pointwise limit is ...
    (sci.math)