Let $X=C[0,1]$ i.e. the set of continuous functions on $[0,1]$ with $L^2$ norm. Then there exists an $F\in X^*$ i.e. dual of $X$ such that there exists no $g \in X$ so that  $$ F(f) = \langle f,g \rangle$$ holds for all $f \in X$.  My effort: Since $(X,|| \cdot ||_{2})$ is not complete. Hence it is not a Hailbert space. We can't apply the Riesz Representation Theorem.  Any help in this regard would be highly appreciated! Thanks in advance!
I wanted to find a counter example to show that the completeness of the inner product space is necessary in . Please give an example of a bounded linear functional $T$ on an incomplete inner product space $X$ which do not have any inner product representation i.e. there does not exist any $z$ in $X$ s.t. $T(x)= \langle ,z\rangle$ for all $x$ in $X$.
Why should I put a ~ before \ref or \cite? My question is in the title of this post, but extends to other referencing commands like \eqref, \citep, \citet (and the like) as well. PrintWriter and PrintStream never throw IOExceptions Some days ago I realized that  (as well as ) never throw an IOException when writing, flushing or closing.  Instead it sets an internal flag (trouble=true) when an error occurs. It's not possible to get the exact exception, but only if there was some exception ().  My question is: why would one want to have such behavior? Isn't that bad API design? What are the minimum content and traffic requirements for Google AdSense? I know in TNG that all computer interfaces are now touch-screen, and that got me thinking: when was the first reference to touch-screen computer interfaces in Sci-Fi made? Linear mixed effect model vs. Ordered Probit vs. Ordered Logit with ordinal response I have a set of data with an ordinal response ranging from 1-5 (worst to best) and a categorical predictor with five unordered levels. The experiment is a language experiment whereby subjects are asked to rate different sentence types. In the literature it seems that people fit lmer() most of the time by scaling the ordinal response per subject (i.e. taking the mean response and standard deviation of that subject and dividing the every single response by it.). On the other hand, it seems a reasonable approach to use ordered linear mixed effects probit models or ordered linear mixed effects logit models as implemented in e.g. MCMCglmm() or clmm().  (1) I find myself asking what the best method would be. When is one method to be prefered over the other in an experiment as I have sketched above?  (2) Why do people prefer fitting an lmer() model with a scaled response? What is the advantage?  (3) If it is advisable to use an ordered probit or logit model, how do I decide between an ordered probit or ordered logit model?  Thanks for any help! Which package to use for presentations? Beamer, Prosper, or Other Finding local maximum and minimum Generalized Tube Lemma build argument lists containing whitespace JavaScript YAML Parser Can the sum of the first $p$ factorials ever be a perfect power for $\ p>3\ $? Has $$\sum_{j=1}^p j!=q^r$$ , where q,p,r are positive integers, and r > 1 , a solution ?   I solved partially, if r is even, then RHS is a perfect square, and there is no doubt in that. Therefore, the left side must be a perfect square as well. But for p>3, the last digit of LHS is 3 which is not the last digit of any square. Hence, p&lt;4, and hence manually checking, only solutions are p =1,3. But i cannot generalize when r is odd.  Any solution to the next part would be helpful! I can't not be pregnant This is a quote from a  where she talks about her miscarriage:     "It was awful," she told the Daily Mail's You magazine. "It happened really late, and I wasn't expecting it at all. "For a long time I just couldn't understand that it had happened and why. I was like 'But I can't not be pregnant. I'm already past the first trimester, and I have three car seats in my head.' To this day I feel like I'm missing that kid. But it wasn't meant to be and you have to trust the universe. "   What does she mean by saying "I can't not be pregnant?" Is this "can't not" construction a grammatical and common usage? Python re.sub with a flag does not replace all occurrences Best Algebraic Geometry text book? (other than Hartshorne) If $\lim_{x \to +\infty} f'(x) = L$, then $\lim_{x \to \infty} \frac {f(x)}{x} = L$ I'm trying to solve this question:     Let $f:[0,+\infty) \to \mathbb{R}$ be derivable and $\lim_{x \to +\infty} f'(x) = L$, then $\lim_{x\to \infty}\frac {f(x)}{x}=L$.   I'm trying to solve this question using l'Hôpital rule, but I couldn't use it, because I don't know if $\lim_{x\to +\infty} f(x)=+\infty$.  I need help.  Thanks a lot. Proof of the L'Hôpital Rule for $\frac{\infty}{\infty}$ I ask for the proof of the L'Hôpital rule for the indeterminate form $\frac{\infty}{\infty}$ utilizing the rule for the form $\frac{0}{0}$.   Theorem: Let $f,g:(a,b)\to \mathbb{R}$ be two differentiable functions such as that:  $\forall x\in(a,b)\ \ g(x)\neq 0\text{ and }g^{\prime}(x)\neq 0$ and $\lim_{x\to a^+}f(x)=\lim_{x\to a^+}g(x)=+\infty$ If the limit $$\lim_{x\to a^+}\frac{f^{\prime}(x)}{g^{\prime}(x)}$$ exists and is finite, then  $$\lim_{x\to a^+}\frac{f(x)}{g(x)}=\lim_{x\to a^+}\frac{f^{\prime}(x)}{g^{\prime}(x)}$$  My attempt:  Since $\lim_{x\to a^+}f(x)=+\infty$, $$\exists \delta&gt;0:a&lt;x&lt;a+\delta&lt;b\Rightarrow f(x)&gt;0\Rightarrow f(x)\neq 0$$  Let $F,G:(a,a+\delta)\to \mathbb{R}$, $F(x)=\frac{1}{f(x)}$, $G(x)=\frac{1}{g(x)}$. Then by the hypothesis $\lim_{x\to a^+}F(x)=\lim_{x\to a^+}G(x)=0$, $$\forall x\in(a,b)\ \ G(x)\neq 0\text{ and }G^{\prime}(x)=-\frac{1}{g^2(x)}g^{\prime}(x)\neq 0$$ The question is, does the limit $$\lim_{x\to a^+}\frac{F^{\prime}(x)}{G^{\prime}(x)}=\lim_{x\to a^+}\frac{-\frac{1}{f^2(x)}f^{\prime}(x)}{-\frac{1}{g^2(x)}g^{\prime}(X)}=\lim_{x\to a^+}\frac{g^2(x)f^{\prime}(x)}{f^2(x)g^{\prime}(x)}$$ exist?   The limit $$\lim_{x\to a^+}\frac{f^{\prime}(x)}{g^{\prime}(x)}$$ exists by the hypothesis but we don't know if the limit $\displaystyle\lim_{x\to a^+}\frac{g^2(x)}{f^2(x)}$ exists to deduce that the limit $$\lim_{x\to a^+}\frac{F^{\prime}(x)}{G^{\prime}(x)}$$ exists to use the L'Hôpital Rule for the form $\frac{0}{0}$.  EDIT: After discussing it with other users in the site, we came to the conclusion that this proof is only partial and can't logically be continued to yield the Theorem.  As a result, the rule for the $\frac{0}{0}$ form can't be used to proove the rule for the  $\frac{\infty}{\infty}$ form. Mr. Tavares and myself have already given two different proofs (with the pretty much the same main idea) of the Theorem in question using Cauchy's Mean Value Theorem. You can read them below. You can also read the proof Rudin gives for a stronger version of the Theorem (that does not suppose that $\lim_{x\to a^+}f(x)=+\infty$) in his book Principle of Meathematical Analysis. If you have any objections in either proofs please let me know. Thank you. Commutator of a group Include the "People Reached" and "Helpful Flags (count)" data in SEDE I know the new profile design hasn't been updated for  on the network yet, so maybe this is a bit too early. Nonetheless, please include the &quot;People Reached&quot; data in  at your earliest convenience. Also, I was just made aware that users helpful flag counts are now publicly displayed on their  towards the top-right, underneath the people reached stat. So please include this data in SEDE as well. Convert ls -l output format to chmod format How to decide whether missing values are MAR, MCAR, or MNAR I have a large proteomics dataset. In the rows I have the proteins , and in the rows I have the samples.The dataset contains a lot of missing values.  I would like to know I can find out whether missing values are MAR, MCAR, or MNAR, and how I can decide the best imputation technique. Kind regards.