Show that ($\mathscr{P}(\Bbb{R})$)' and $\Bbb{R}^\infty$ are isomorphic. Apologies for any bad formatting. This is number 35 from chapter 3.F of Axler's Linear Algebra Done Right, if you have the text to view. I've been stuck on this one for quite some time: obviously because the vector spaces are infinite dimensional, we cannot create the isomorphism using bases of the spaces. Also, because $\Bbb{R}^\infty$ contains infinite sequences, it is not clear to me how to create an injective function... I also thought that maybe I could develop the isomorphism using finite dimensional subspaces of the respective spaces, which is relatively easy, but being able to abstract that iso for the infinite dimensional spaces is beyond me. Any assistance/explanation would be greatly appreciated.
Show that the dual space of the vector space of all polynomials is isomorphic to the infinite-dimensional Euclidean vector space over the reals May I please ask how to show the dual space of the vector space of all polynomials is isomorphic to the infinite-dimensional Euclidean vector space over the reals?  (i.e. Show that $(\Bbb{R}[X])^*$ and $\Bbb{R}^{∞}$ are isomorphic where $\Bbb{R}$ means the set of all reals)
How do I format a backslash as code with backticks in comments? I'm trying to write a single backslash in a comment as code formatted by backtick. Like this:  `\`   but it escapes the backtick and doesn't close the "inline code block".  I've read  but couldn't figure out.  How do I format a single backslash in comment as code? UEFI machine doesn't boot Ubuntu through NVRAM bootcatalog. How to fix? Update to latest version of java after ppa is discontinued If I pay off lease amount early for a car, is there any impact on my credit?  Let's say I'm paying $400/month for 36 months, which is 14,400 total. If I pay 14,400 within 6 months, are there any disadvantages on my credit? Is my way of solving and my answer correct?  Let $f(x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^5}{5}$   And $g(x)=f^{-1}(x)$  Find $g'''(0)$  My attempt:  We know that $g'(x)=\frac{1}{f'(g(x))}$  $f'(x)=1+x+x^2+x^3+x^4=\frac{x^5-1}{x-1}$  $f'(g(x))=\frac{(g(x))^5-1}{(g(x))-1}$  $\Rightarrow g'(x)=\frac{(g(x))-1}{(g(x))^5-1}$  $\Rightarrow g'(x)[(g(x))^5-1]=g(x)-1$  Similarly differentiating again,  $\Rightarrow g'(x)[5(g(x))^4]+[(g(x))^5-1]g''(x)=g'(x)$  Similarly differentiating again,  $\Rightarrow g'(x)[20(g(x))^3g'(x)]+[5(g(x))^4]g''(x)+[(g(x))^5-1]g'''(x)+g''(x)[5(g(x))^4]=g''(x)$  Putting $x=0$,  $g(0)=f^{-1}(0)=0$  $g'(0)=\frac{1}{f'(g(0))}=\frac{1}{f'(0)}=1$  similarly,$g''(0)=-1$  $\Rightarrow g'''(0)=1$ Familiar form of address for a young, subordinate, woman that connotes respect (Female equivalent to 'Son') A(n often male) paternal figure could use the term 'son' in a fatherly way without referring to his biological Son.  Imagine a man has been verbally abused by a customer at work. His manger might say to him:     Come here son, I have something to tell you   about rude customers   Emphasis on the word I want gender-swapped  Is there a feminine equivalent for a subordinate that you respect?   From      son  (sŭn)   n.    1. One's male child.    2. A male descendant.    3. A man considered as if in a relationship of child to parent: a son of the soil.    4. One personified or regarded as a male descendant.    5. Used as a familiar form of address for a young man.    6. Son Christianity The second person of the Trinity.     Is there an equivalent for talking to a woman in a fatherly way?  The equivalent for TFD entry for daughter does not qualify it as 'familiar'        (often capital) a form of address for a girl or woman      Further more often such 'father-daughter' language patterns are negative, dismissive and lack respect. How do I get the  value of a character as an int in ? Degree of the extension $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ over $\mathbb{Q}$ Following my  the book then asks      Use this to determine when the field extension  $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ over $\mathbb{Q}$  is biquadratic (where $a,b\in\mathbb{Q}$)   "this" means what proved in the linked post.  My thoughts: First, I don't know what to say when $a^2-b$ is not a square. When $a^2-b$ is a square I think that the extension is of degree $2$ if $a=b$, this leaves the case $a\neq b$ which I am also having problems with and I don't know what to do (I have some cases like if $m,n$ are different primes then the degree of the extension is $4$, but I don't have something that is general)  Can someone please help my do this exercice  ? I'm a software developer and as such I build and run programs hundreds of times per day. I've noticed that since I started using Catalina, after compiling and linking there is a delay (which can be seconds) when the internet is slow or having problems (I have bad internet often). If I disable wifi and rebuild, the delay goes away. How can I disable this so my builds run faster? Is there any way I can see at what time the commands were executed from the bash history? We can see the order but is there any way I can get the time also?  Bottom-Line: Execution time in the Bash history Proofs of Determinants of Block matrices Tourist to F1 change of status within 60 days I have a PhD place at a US university starting Aug 21 but want to visit a different university for casual research for 6 weeks from July 1. This is longer than the 30 day grace period for the F1. Can I arrive on the visa waiver program or tourist visa in July and then just stay on for the PhD? I guess I might need to change status. Will this work?  Please don’t suggest exiting and reentering, that’s too expensive.  (British masters student in UK) How do "and" and "or" act with non-boolean values? How to convert light activated switch According to my text book:  The Lorentz Force F on a charged particle moving in an electromagnetic field is given by:  $$F= qE + q(v\times B)$$  then it states that "it is to be pointed out that only the electric force does work, while the no work is done by the magnetic force which is simply a deflecting force."  How can this by true won't the the magnetic force have a considerable magnitude, will it now impart kinetic energy to the charge? Any explanations will help a lot. Please keep the terminology simple, thanks. Can I enter the Schengen area with a German long-term visa through a different country? I am an Indian national. Right now I am staying in Israel. I will be moving to Germany on the last week of this month with german national visa directly from tel aviv. Ticket price of direct flight is high. So I am thinking to buy either Air France or LOT for travelling having layover at Paris or Poland. Am I allowed to do that? I mean is it mandatory to enter in Germany first? May I travel with these airlines with proper transit visa? Finding acute triangle probability A stick is broken in three parts of any length ,  if we join the pieces then what's the probability of getting a acute angled triangle? Why is the potential energy equal to the negative integral of a force? Why is the potential energy equals to the negative integral of a force? I am really confused with this negative sign. For example, why there is a negative sign in the gravitational potential energy and what does it mean?   I read that the negative sign means you do the same force but in the opposite direction.  Doesn't that mean the object shouldn't move? Is this algebraic identity obvious? $\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1$ If $\lambda_1,\dots,\lambda_n$ are distinct positive real numbers, then  $$\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1.$$ This identity follows from a probability calculation that you can find at the  top of page 311 in the 10th edition of Introduction to Probability Models by Sheldon Ross.   Is there a slick or obvious explanation for this identity?   This  question is sort of similar to ; clearly algebra is not my strong suit! Which manga chapter correspond to the end of episode 12 of season 2 of the anime?