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Let $f:[0,1] \to \mathbb R$ be a continuous function and the sequences $(a_n)_n,(b_n)_n$ s.t. $$\lim_{n\to \infty} \int_0^1 |f(x)-a_nx-b_n| dx=0.$$ Prove that $(a_n)_n,(b_n)_n$ are convergent.

I know that $$\left|\int_0^1f(x)dx\right| \le \int_0^1|f(x)|dx.$$ Can somebody help me, please?

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Consider the metric space $E$ of the continuous functions on $[0,1]$, with the distance $d : E \times E \rightarrow \mathbb{R}_+$ defined by $$d(f,g)=\int_0^1 |f(t)-g(t)| \mathrm{dt}$$

Consider the subspace $A$ of affine functions.

Consider the application $\varphi : [0,1]^2 \rightarrow E$ defined for all $(a,b) \in [0,1]^2$ by $$\varphi(a,b) = \lbrace f : t \rightarrow (b-a)t+a \rbrace$$

Then it is easy to see that $\varphi$ is continuous. Moreover, its image is precisely $A$. You deduce that $A$ is a compact subset of $E$ (as a continuous image of the compact $[0,1]^2$), and therefore $A$ is closed.

This proves, in the case of your question, that such a function $f$ has to be affine, as a limit of the closed set of affine functions.

So there exists $a,b$ such that $f(x)=ax+b$. You can rewrite the hypothesis $$\int_0^1 |(a-a_n)x +(b-b_n)| \mathrm{dx} \rightarrow 0$$

It is easy now to prove that the only possibility is that $a_n \rightarrow a$ and $b_n \rightarrow b$.

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    $\begingroup$ Very ingenious argument. +1 $\endgroup$ – Paramanand Singh Mar 8 at 17:31
  • $\begingroup$ Don't you know a simpler proof?I do not know metric spaces. $\endgroup$ – Gaboru Mar 8 at 20:29
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Here is an elementary proof:

Lemma

If $\int_0^{1} |f(x)-f_n(x)|dx \to 0$ (with $f,f_1,f_2,...$ continuous) and $F(x)=\int_0^{x} f(y)dy,F_n(x)=\int_0^{x} f_n(y)dy$ then $\int_0^{1} |F(x)-F_n(x)|dx \to 0$.

Proof of lemma: $$\int_0^{1} |F(x)-F_n(x)|dx \leq \int_0^{1}\int_0^{x} |f(y)-f_n(y)|dy dx$$ $$ =\int_0^{1}\int_y^{1} |f(y)-f_n(y)|dxdy =\int_0^{1} (1-y)|f(y)-f_n(y)|dy \leq \int_0^{1} |f(y)-f_n(y)|dy$$. This proves the lemma.

Now the hypothesis implies $\int_0^{1} (a_nx+b_n) dx$ converges and the lemma shows that $\int_0^{1} (a_nx^{2}/2+b_nx) dx$ also converges. From these two it is quite easy to show that $(a_n)$ and $(b_n)$ both converge.

Another proof using basic measure theory. It is given that $a_nx+b_n \to f$ in $L^{1}$. This implies there is a subsequence which converges almost everywhere, say $a_{n_k}x+b_{n_k} \to f(x)$ almost everywhere. Form this (using two values of $x$ for which convergence holds) we see that $a_{n_k}$ and $b_{n_k}$ both converge, say to $a$ and $b$ so $f(x)=ax+b$ for some $a$ and $b$. Note that this last equation uniquely determines $a$ and $b$. In fact $b=f(0)$ and $a =f(1)-f(0)$ We can now argue that every subsequence of $a_{n_k}$ has a subsequence converging to the same limit, so $\{a_n\}$ is convergent. Similarly $\{b_n\}$ is convegent.

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  • $\begingroup$ Why are the sequences $(a_n) $ and $(b_n) $ convergent? $\endgroup$ – Fred Mar 8 at 21:21
  • $\begingroup$ A sequence $(x_n)$ in a metric space converges to $x$ iff every subsequence of $(x_n)$ has a further subseqeunce which converges to $x$. @Fred $\endgroup$ – Kavi Rama Murthy Mar 8 at 23:17
  • $\begingroup$ @Gaboru I have added another elementary proof. $\endgroup$ – Kavi Rama Murthy Mar 8 at 23:37
  • $\begingroup$ Mister, do you know how to proceed Fred's argument? $\endgroup$ – Gaboru Mar 9 at 17:06
  • $\begingroup$ @Gaboru Fred's answer is incomplete and I have completed it by considering anti-derivatives of $f$ and $a_nx+b_n$. $\endgroup$ – Kavi Rama Murthy Mar 9 at 23:13
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We have $|\int_0^1 (f(x)-a_nx-b_n) dx| \le \int_0^1 |f(x)-a_nx-b_n| dx.$

Hence $\int_0^1 (f(x)-a_nx-b_n) dx \to 0$.

Since $\int_0^1 (f(x)-a_nx-b_n) dx = \int_0^1 f(x) dx-\frac{1}{2}a_n-b_n$, we see:

$\frac{1}{2}a_n+b_n \to \int_0^1 f(x) dx$.

Can you proceed ?

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    $\begingroup$ But that doesn't imply that both of them are convergent. Because we can be in a infinity-infinity case $\endgroup$ – Gaboru Mar 8 at 11:58
  • $\begingroup$ I managed to get that alone. I don't know how to proceed. $\endgroup$ – Gaboru Mar 9 at 17:04
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Let be $L_n(x) = a_n x + b_n$. The hypothesis says that $L_n\to f$ in $L^1$. Consider the vector subspace generated by the polynomials of degree $\le 1$ and $f$. Obviously, is finite-dimensional. But in finite dimension all the norms are equivalent, so the convergence $L_n\to f$ is uniform...

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