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2d
comment How to show without calculator that $\left\lfloor\, \log_{10}{999^{999}}\right\rfloor =\left\lfloor\, \log_{10}{999^{999}}+\log_{10}2\right\rfloor$
@Swapnil No, you also have to show that $\log_{10} 999^{999}$ has a distance greater than $\log_{10} 2$ to the next larger integer. Consider e.g. $0 = \lfloor 0.9 \rfloor \neq \lfloor 0.9 + 0.2 \rfloor = 1$.
2d
comment How to show without calculator that $\left\lfloor\, \log_{10}{999^{999}}\right\rfloor =\left\lfloor\, \log_{10}{999^{999}}+\log_{10}2\right\rfloor$
@Swapnil $\lfloor \cdot \rfloor$ is the floor function, i.e. $\lfloor x \rfloor$ is the largest integer which is smaller or equal to $x$, e.g. $\lfloor 1.2 \rfloor = \lfloor 1.9 \rfloor = 1$.
2d
revised Question about Convergence Definition for Finite Difference Scheme
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2d
answered Question about Convergence Definition for Finite Difference Scheme
2d
revised Equivalent Solovec norms (atypical)
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2d
answered Equivalent Solovec norms (atypical)
Jan
18
awarded  Yearling
Jan
18
comment Prove that L 2 PC[−1, 1] is not a complete normed space
The steps are disjoint and the step height doesn't depend on $n$, so for $I_k = [1-2^{-k+1}, 1-2^{-k})$ there holds $f_n|_{I_k} = 2^{-k}$ for all $n \geq k$. The same is true for the limit, so $f(x) \neq 0$ for $x \neq 1$. The step heights tend to zero for $k \to 0$, but this only means that $f(x) \to 0$ for $x \to 1$.
Jan
18
answered Prove that L 2 PC[−1, 1] is not a complete normed space
Jan
10
comment Convergence in dual of Sobolev space
Most spacial discretization techniques I know need a Hilbert space structure, so I work with $p=2$ most of the time. That doesn't mean that $L^1$ or $L^\infty$ are uncommon though. This whole discussion is moving a bit off-topic, feel free to open a chat room if you have further questions.
Jan
9
awarded  Commentator
Jan
9
comment Convergence in dual of Sobolev space
My first approach would have been the elementary one, but the whole Hölder estimation argument is nothing more than $(L^p(Ω))^* \cong L^{p'}(Ω)$, so in retrospective I would consider your approach actually to be the more beautiful. If you wouldn't write out all the objects. :) I work with PDEs from a numerical point of view, so Sobolev spaces turn up everywhere.
Jan
9
revised Convergence in dual of Sobolev space
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Jan
9
comment Convergence in dual of Sobolev space
I added more details to my response. I meant Cauchy-Schwarz for $\mathbb R^n$, I probably shouldn't have mentioned that explicitly. :)
Jan
9
revised Convergence in dual of Sobolev space
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Jan
9
answered Convergence in dual of Sobolev space
Jan
9
awarded  Editor
Jan
9
revised Question about continuous from right or left
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Jan
9
answered Question about continuous from right or left
Jan
7
answered The Sobolev Space $H^{1/2}$