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3

I'm not in this area but I can say you that the main issue here is the application of this kind of Mathemathics to Quantum Mechanics. Indeed, even if Hilbert didn't started studying the argument with this in mind it was soon finded that this branch of mathemathics was really suitable to modelize Quantum phenomenas. Indeed what happened is that soon after ...


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For a given $p\gt1$, consider $$ a_{n,k}=\left\{\begin{array}{cl} \dfrac1{n^{1/p}}&\text{if }1\le k\le n\\ 0&\text{if }k\gt n \end{array}\right. $$ Then $$ \begin{align} \left(\sum_{k=1}^\infty a_{n,k}^p\right)^{1/p} &=\left(\sum_{k=1}^n\frac1n\right)^{1/p}\\[6pt] &=1 \end{align} $$ while $$ \begin{align} \sum_{k=1}^\infty a_{n,k} ...


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Partial answer, for $p>2.$ Let $P_n(x)=\sum_{j=1}^n j x^j .$ We have $$\|P_n\|^p=\sum_{j=1}^n j^p<\sum_{j=1} ^n\int_j^{j+1}y^p dy=\int_1^{n+1}y^p dy=$$ $$=\frac{(n+1)^{1+p}-1}{1+p}<\frac {(n+1)^{1+p} }{1+p}.$$ $$\text {So }\; \|P_n\|<\frac {(1+n)^{(1+1/p)}}{(1+p)^{1/p}}\;\text { But }\; P_n(1)=(n^2-n)/2.$$ Let $Q_n=P_n/n^{1+2/p}.\;$ Then ...


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Its always a bit hard to guess what another person might find intuitive, but here are my two cents on the topic. You can interpretate the elements of $\mathbb{R}^n$ as functions from the set $\{1,...,n\}$ to $\mathbb{R}$, where for $f \in \mathbb{R}^{n}$, $f(i)$ would just be the $i$-th component of the vector. We know from linear algebra that any linear ...


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Simple example of two non-equivalent norms on infinietly-dimensional space: Consider space of all contnuously differentiable functions $X = C^1 [0,1]$. Then equipping it with the norm: $$ \|f \|_{C^1} = \sup \limits_{x \in [0,1]} |f| + \sup _{x \in [0,1]} |f'| $$ gives us a complete space (Banach space), but if we consider norm: $$ \|f \|_\infty = \sup ...



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