- #1

Sudharaka

Gold Member

MHB

- 1,568

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Here's a question I encountered and I need your help to solve it.

**Question:**

Let \(V\) be the space of real polynomials of degree \(\leq n\).

a) Check that setting \(\left(f(x),\,g(x)\right)=\int_{0}^{1}f(x)g(x)\,dx\) turns \(V\) to a Euclidean space.

b) If \(n=1\), find the distance from \(f(x)=1\) to the linear span \(U=<x>\).

**My Answer:**

In our notes it's given that an Euclidean space is a pair \((V,\,f)\) where \(V\) is a vector space over \(\mathbb{R}\) and \(f:V\times V\rightarrow\mathbb{R}\) is a positive symmetric bilinear function. So therefore I thought that we have to check whether the given bilinear function is positive symmetric. It's clearly symmetric as interchanging \(f\) and \(g\) won't matter. But to make it positive we shall find a condition. Let,

\[f(x)=a_{0}+a_{1}x+\cdots+a_{n}x^{n}\]

\[g(x)=b_{0}+b_{1}x\cdots+b_{n}x^{n}\]

Then,

\[f(x)g(x)=\sum_{k=0}^{n}\left(\sum_{i=0}^{n}a_{i}b_{k-i}\right)x^{k}\]

Hence we have,

\[\left(f(x),\,g(x)\right)=\int_{0}^{1}f(x)g(x)\,dx>0\]

\[\Rightarrow \sum_{k=0}^{n}\left(\sum_{i=0}^{n}a_{i}b_{k-i}\right)\frac{1}{k+1}>0\]

Is this the condition that we have to obtain in order for \(V\) to become an Euclidean space?