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I have the given space with norm:

Let $$E=C[a,b],$$ $$||x||=\underset{a\le t\le b}{\max}|x(t)|$$

I make the example of two elements in the dual space $E^*$:

$x_\alpha(t)=\sin mt$ and $x_\beta(t)=\sin nt$ on $[-\pi,\pi]$. Being an Euclidean dual space, their scalar product is:

$$2\int_{-\pi}^\pi \sin nt\sin mt dt=\cos\big((m-n)t\big)-\cos\big((m+n)t\big)$$ where the integral of the product of the two sine functions vanishes. This satisfies that the inner product must be zero for orthogonal functions.

However, when I want to prove this by using their norm, I get a contradiction.

$$||\sin mt+\sin nt||=1+1=2,$$ however $$||\sin mt-\sin nt||=1-1=0$$

Therefore, $$||\sin mt+\sin nt||\ne||\sin mt-\sin nt||,$$ so it cannot be an inner product space.

So if $C[a,b]$ is not an inner product space, then these two elements cannot be examples of elements in $E^*$!

Is there an error here, in the estimation of the norm?

I am not sure,

Any hints appreciated

Thanks

Gary
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Luthier415Hz
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    I'm afraid what you are writing does not make a lot of sense. The functions you mention are not elements of the dual space of E, and I don't see why you would know the dual space is an inner product space. Because of this, your following computations don't give any insight to the original question. To prove or disprove it being an inner product space, consider using the Parallelogram Law, see https://math.stackexchange.com/questions/21792/norms-induced-by-inner-products-and-the-parallelogram-law?rq=1 – Richard Jensen Dec 14 '22 at 10:56
  • The dual space of an Euclidean space has inner product too. This is why I included that. – Luthier415Hz Dec 14 '22 at 10:59
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    Okay, but $C[a,b]$ with with that inner product (the integral of their product) does not induce the norm you are talking about here (the so called "supremum" or "maximum" norm) – Richard Jensen Dec 14 '22 at 11:34
  • You are right! I was confused by the nomenclature "E" , which is normally used for Euclidean spaces. So this is really a simple space of continuous functions on a closed interval. Therefore, I can choose any linear function as an element, which converges in the metric. An example would be the Power series of $e^x$ for example – Luthier415Hz Dec 14 '22 at 11:42

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