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Jul
26
comment The Star Trek Problem in Williams's Book
Why not try the problem in the one-dimensional setting and see what happens?
Jul
19
awarded  Critic
Jul
4
comment History of Dual Spaces and Linear Functionals
@MattE I like that explanation. In fact, that is exactly how it was done historically according to the stuff I have read.
Jul
4
comment Dimension of a cyclic submodule of a finite group representation
Isn't it true that the span of $\{\rho_g v : g \in G, v \in V\}$ is equal to the span of $\{\rho_g v : g \in G, v \in B\}$ where $B$ is any basis of $V$? If $G$ is finite, then you can compute the dimension of this latter span by checking for linear independence of the vectors in $\{\rho_g v : g \in G, v \in B\}$.
Jul
4
comment Bound on $E[X Y]$, where $Y$ is bounded in terms of $X$
If $X \ge Y^k$, then $XY \ge Y^{k+1}$ and so $E[XY] \ge E[Y^{k+1}]$. I don't know what to do with $E[Y^{k+1}]$ though.
Jul
3
comment History of Dual Spaces and Linear Functionals
Nice. I will definitely have to read that book.
Jul
3
answered History of Dual Spaces and Linear Functionals
Jul
2
comment History of Dual Spaces and Linear Functionals
I disagree that the concept of a linear functional occurs "everywhere". The two examples you gave are well and good but they will not force me to start thinking about linear functionals in general. I am wondering what Riesz's motivation was in producing the RRT. You mention that the RRT can be used to define the cross product using determinants. I find that interesting. Do you have a reference?
Jul
2
comment History of Dual Spaces and Linear Functionals
If I were exploring the difference between row vectors and column vectors, I would make two observations: row vector x column vector = scalar and column vector x row vector = matrix. That this would lead me to the concepts of linear functional or dual space seems unlikely to me.
Jul
1
asked History of Dual Spaces and Linear Functionals
Jun
13
comment Union of compact sets in a convergence space
Ah, of course! Thanks a lot.
Jun
13
accepted Union of compact sets in a convergence space
Jun
13
comment Union of compact sets in a convergence space
This is true if the convergence spaces are topological by the argument you just gave. The general case though...
Jun
13
asked Union of compact sets in a convergence space
Jun
8
awarded  Caucus
Jun
1
comment Continuous and Open maps
A continuous function that maps open sets to open sets is just called an open map as far as I know.
Apr
20
comment Is there a way of defining the notion of a variable mathematically?
Thanks for the example. However, I am not convinced this clarifies what a variable is. Calling the symbol $x$ a variable when defining a polynomial ring $F[x]$ does not define what a variable is.
Apr
20
accepted Is there a way of defining the notion of a variable mathematically?
Apr
19
comment The topology of distributions
Excellent. Thanks.
Apr
19
comment Is there a way of defining the notion of a variable mathematically?
I do not see the connection. Can you give me a concrete example?