@Thorgott Hm, if $W$ isn't complete there's a Cauchy sequence $\{w_n\}$ with no limit. If $V$ is a Hilbert space, you could fix an arbitrary vector $v \in V$, look at the projection $V \to \langle v \rangle$ and compose that with $\langle v \rangle \to W$, sending $v$ to $w_n$. That gives a sequence $T_n : V \to W$ of linear maps which doesn't converge in the operator norm, right?
user131753
@SubhasisBiswas Intuition is always misleading unless you have a rigorous formulation of that in purely formal terms.
If it did converge to $T$, then $\|T(v) - w_n\| = \|T(v) - T_n(v)\| \leq \|T - T_n\|_{op} \cdot \|v\|$, which goes to zero. That'd force $w_n$ to converge to $T(v)$.
And I mean $\{T_n\}$ is Cauchy as well, because it kills everything orthogonal to $v$ and $\{T_n(v)\}$ is Cauchy in $W$.
user131753
4:22 PM
@Thorgott I am pretty much sure that this theorem follows from Hahn-Banach Theorem in a non-trivial way.
The orthogonal projection of an inner product space $V$ to a closed subspace $U$ is usually given by the minimum distance to $U$, which needs continuity arguments
I think you end up using completeness crucially there
But I could be wrong, not too familiar with functional analysis
I don't know functional analysis, but checking the reference I have at hand, it seems that being a Hilbert space is at least a sufficient condition for the existence of orthogonal projections on non-empty, closed subspaces. This leaves the case of non-Hilbert $V$ and infinite-dimensional $W$; the question might be more non-trivial than anticipated...
Hey I have a simple question about cosets that is evading me if anybody can provide a hand. Is it true that for any subgroup $H$ of a finite group $G$, $\exists x \in G$ such that $G = \bigcup\limits_{i=0}^{|H|-1}x^iH$?
Supposing that there could indeed be a possibility that $G$ contains $4$ distinct elements $\{x,y,z,w\} (=B)$ of order $3$:
Let $A=\{x, x^2, e\}$ be the cyclic subgroup generated by $x$ Then $[G:H]=2$, which further implies that $x^2 \in H, \forall x\in G$.
Now, $y^2, y \in A$. It must be the case that $y^2=x$ and $y=x^2$ (otherwise, $x=y$, or $y=e$, an impossible case by our assumption). Now, there is another element $z$ of order $3$. Again, $z, z^2 \in A$. So, $z=y^2$ and $z^2=y$. But, this in turn implies that $z=x$. So, the elements in $B$ fail to be distinct. So, $|B|=2$ since, $|B|…
The exercise that the proof is for is from Gallian's Contemporary Abstract Algebra Ch7E38. It's assumed at the beginning of the proof to derive most of the results. The exercise as stated is "Prove that if G is a finite group, the index of its center cannot be prime."
I'm doing self-study so I don't have classmates/professor to ask. :\
@BalarkaSen Actually, I think there is no difficulty in projecting orthogonally on a finite-dimensional subspace. Fix $v_0\in V$ s.t. $\lVert v_0\rVert=1$. Then $v\mapsto\langle v,v_0\rangle v_0$ is an orthogonal projection on $\langle v_0\rangle$. I checked the conditions and either I'm missing a very subtle detail in the case that $V$ is infinite-dimensional or this works and gets rid of the completeness assumption.
@SubhasisBiswas well no 'cuz then $G$ would have to contain a generator and thus be cyclic but not all finite groups are cyclic so the statement becomes false
I've always loved helping people. That's why I have chosen a career in imposing the will of politicians with the threat of violence. That's why I am proud to be a police officer
if your don't have personality traits that enable you to receive respec by default don't worry, the government have a wide range of costumes that force people to respect everything you say about all the things that are rely good (Jesus, money, fossilfuels) and all the things that are banned
Hey all. I'm trying to figure out how to calculate a y value for a sine wave based on 25 x values between 0 and 1 (so .04 steps). I am going with the assumption that the sine wave will have a maximum magnitude of 1 as well.
@BrandenBoucher: You'll need to explain your question more carefully. What do you mean by "y value for a sine wave based on 25 x values"? You have numerical data for every integer multiple of $x=.04$? So can you predict the period (or frequency) if you know it has to be a sine wave?
@TedShifrin well I'm just trying to calculate points along a cosine starting at 0,0.5 and ending at 1,0.5 with an amplitude of 1 (so a min value of -0.5 for y)
Where did cosine come from? Where did the amplitude of 1/2 come from? And how in the world do we know that we get one wave as $x$ goes from $0$ to $25$?
Context: we left Minneapolis about an hour or so before sunset. So while flying east, the sun dipped below the horizon with that characteristic red glow above it
It's not just that we are doing things like making abortions completely illegal, the idiots that align themselves with their political party have become so triballistic that there is no acceptable middle ground.
@SubhasisBiswas: I think the interesting conjecture is this. If the condition holds (with the typo corrected), must $H$ be a normal subgroup (so that, of course, $G/H$ is then a cyclic group)?
@BrandenBoucher: You won't be surprised to know that for me most of the Democratic party is too far to the center. But the Republican party has totally sold its soul and has long since ceased to play by the Constitution.
I want to set up a stack exchange community for maple users but it says error no user access for the area 51 one thingy so I guess there was no way
what's the url for whatever stackexchange is in canada
and why does stack exchange plug Nellis air force base hoha sillys anyway I mean unless they have q clearance which would make sense why the so many non human acting nimrods exisit but yes
Could someone please help me with a thing? I want to calculate something but I am not sure how to do that. It's, for a game. The player has a current health ($Hc$) and maximum health ($Hm$) numbers and I want to apply a special damage reduction ($R$). This damage reduction increases inversely to its remaining health percentage ($\frac{Hc}{Hm}$). The formula of this damage reduction is: $F(x) = (10+e)/e^{-x*10}/100$, where $x = \frac{Hc}{Hm}$, this reduction is multiplied to the incoming raw damage ($D$) and the result if the reduction of health.
Oh, I forgot a dollar symbol and so the latex became mad. This is the fixed latex: $Hc = 30$ $Hm = 100$ $D = 10$ $R = (10+e)/e^{-(\frac{35}{100})*10}/100 = 0.6459...$ $D_2 = D * R = 6.45...$ $Hc_2 = Hc - D_2 = 23.54...$
With this
$Hc = 30$ $Hm = 100$ $D = 5$ $R = (10+e)/e^{-(\frac{35}{100})*10}/100 = 0.6459...$ $D_2 = D * R = 3.22...$ $Hc_2 = Hc - D_2 = 26.77...$
And (the same but using the current health $Hc = Hc_2$)
My current aproach is very clumpy. Instead of using a formula I did:
Hc = 30
Hm = 100
D = 5
for N in {D, D-1, D-2, D-3... D-(D-1)}
R = (10+e)/e^{-(\frac{Hc}{Hm})*10}/100
Hc = Hc - (1 * R)
Why the complex differentiability of $f(z)$ and real differentiability of $f(z)=f(x,y)$ are two different concept? $H:\mathbb C\to \mathbb R^2 : H(z=x+iy)=(x,y)$ is homeomorphism as well as homomorphism. right?
