- #1

geoffrey159

- 535

- 72

## Homework Statement

Prove that the following families are free in the vector space of continuous mappings from ##\mathbb{R} \rightarrow \mathbb{R}##, with real scalars :

1 - ## (f_\lambda)_{\lambda \in \mathbb{R}^+} : f_\lambda(x) = \cos(\lambda x) ##

2 - ## (f_\lambda)_{\lambda \in \mathbb{R}} : f_\lambda(x) = |x-\lambda| ##

## Homework Equations

## The Attempt at a Solution

Ok, I'm not used to that kind of exercise so it may be all wrong, and I've worked this problem long enough so that it may be even worse than that.

My approach will be to show that any finite sub-family of ##(f_\lambda)_{\lambda\in \mathbb{R}}## is free.

Let ##\lambda_1 < ... < \lambda_N ## be distinct lambdas, and ##(x_1,...,x_N)## be real scalars such that ##\sum_{i = 1}^N x_i f_{\lambda_i}(x) = 0##, for any real ##x##. I want to show that ##x_1 = ... = x_N = 0##

1 -

In this case, ##0\le \lambda_1 < ... < \lambda_N ##. Since one can derivate infinitely many times the function ##f_\lambda##, then for any ##k\ge 0##,

## 0 = \frac{d^{2k}}{dx^{2k}}(\sum_{i = 1}^N x_i f_{\lambda_i}) = \sum_{i = 1}^N x_i \frac{d^{2k}}{dx^{2k}}(f_{\lambda_i}) = (-1)^k \sum_{i = 1}^N x_i \lambda_i^{2k} f_{\lambda_i}##.

The ##(-1)^k## can be overlooked since the whole term is equal to zero. Taking ## x = 0 ##, then ## 0 =\sum_{i = 1}^N x_i \lambda_i^{2k} ##.

So ## x_N = - \sum_{i = 1}^N x_i (\frac{\lambda_i}{\lambda_N})^{2k} ##. Since ##0 \le

(\frac{\lambda_i}{\lambda_N})^{2k} < 1 ## for any ##k##, then ##x_N = 0 ## by taking the limit as ##k## tends to infinity.

Repeating this process, I find ## x_N = ... = x_2 = 0 ##. Now if ##\lambda_1 = 0##, then

##0 = \sum_{i = 1}^N x_i f_{\lambda_i}(x) = x_1 f_{\lambda_1}(x) = x_1 ##. If ##\lambda_1 \neq 0 ##, then ## 0 = \sum_{i = 1}^N x_i \lambda_i^{2k} = x_1 \lambda_1^{2k}##, so ## x_1 = 0## too !

2 - I haven't reached to the conclusion yet.

What I can show is that for any ## x > \max(0,\lambda_N)##, then ## x \sum_{i = 1}^N x_i = \sum_{i = 1}^N x_i \lambda_i ##.

Dividing by ##x ## on both side (legal because x > 0) and taking the limit as ##x## goes to infinity,

## 0 = \sum_{i = 1}^N x_i = \sum_{i = 1}^N x_i \lambda_i ##. I need help for this one