I have a normed space: $$l_p = \{ (x_n)_{n = 1}^{\infty}: \sum\limits_{n = 1}^{\infty}|x_n|^{p} < \infty \}$$ With norm: $$||x|| = (\sum\limits_{n = 1}^{\infty} |x_n|^p)^{1/p}$$ where $p = \dfrac{5}{4}$
And I want to find norm of following functional:
$$f(x) = -x_{1} + x_{100} + \sum\limits_{n = 1}^\infty \dfrac{x_{2n}}{n!}$$
I was try apply Hölder's inequality, but not shure it was right:
$$|f(x)| \le ||x||_p ||g||_q, q = 5$$ and $g$ looks like:
$$g(x) = (-1,\dfrac{x_2}{1!}, 0, \dfrac{x_4}{2!}, 0, \ldots, \dfrac{(50! + 1) x_{100}}{50!}, 0, \dfrac{x_{101}}{51!}, \ldots)$$
But I have no idea how to move forward.