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I am part of a functional analysis course and we are expected to prove the following:

$$\lVert x \rVert _\infty = \lim _{p \to \infty} \lVert x \rVert _p$$ where $$\lVert x \rVert _p = (|x_1|^p + |x_2|^p + \dots + |x_n|^p)^{1/p}$$

I am not expecting an answer for the proof, but need some clarification about the question. Isn't $\lVert x \rVert _\infty$ defined to be $\lim _{p \to \infty} \lVert x \rVert _p$? If not, what does $\lVert x \rVert _\infty$ mean?

Ѕᴀᴀᴅ
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hayev10063
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    The infinity norm is $|x|\infty = \max_i |x_i|$; the definition can be found on [Wikipedia](https://en.wikipedia.org/wiki/Norm(mathematics)#p-norm) for example. As it turns out, it is the limit of the $p$-norms, as $p \to \infty$, see here. – Theo Bendit Oct 14 '22 at 01:22
  • The question makes a lot more sense. Thanks! – hayev10063 Oct 14 '22 at 01:24
  • No problem! It's weird that you haven't seen that defined explicitly in your course before. – Theo Bendit Oct 14 '22 at 01:26
  • Yes, not sure why the professor did not cover it; however, it is my mistake for starting the assignment late and not leaving enough time to ask questions to the professor or TA. – hayev10063 Oct 14 '22 at 01:28

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Over $\mathbb{K}^n,\ \text{the norm }\|.\|_\infty$ is defined by $\|(x_1,x_2,\dots, x_n)\|_\infty= \max\{|x_i|\}_{1\leq i\leq n}$

NotaChoice
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