Question

Which of the following is an inner product in P2? Explain

(a) $<f,g>\mathbf{=}\mathit{f}\left(1\right)\mathit{g}\left(1\right)\mathbf{+}\mathit{f}\left(2\right)\mathit{g}\left(2\right)$

(b)

Short answer

Second one is an inner product.

Step-by-step solution

Consider for part (a). (a)

It needs to check whether the given functions are an inner product or not.

This is not an inner product. It will be an inner product if and only if the functions satisfy the following:

$f\left(1\right)\ne 0,f\left(2\right)\ne 0$

For example, consider below.

Thus, it is not an inner product.

Consider the properties below. (b)

Here it is claimed that it is an inner product. So, it will check all the properties:

Let $f,g,h\in P_2$and $a\in R$.

Check the all properties:

1)

2)

3)

4)

Let f be a non-zero-degree 2 polynomial.

role="math" localid="1660738388720"

The last expression is always positive. Since f is a second non-zero polynomial.

Therefore, it gives $f{\left(1\right)}^2+f{\left(2\right)}^2+f{\left(3\right)}^2>0$. It satisfies all conditions for inner product.

Hence, first one is not an inner product whereas the second one is inner product.