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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?
Apr
19
comment Is there a way of defining the notion of a variable mathematically?
I think I understand what you are getting at. If we take an informal expression with variables and make it formal ala metamath, we would get some string of symbols involving those variables, which is just a string in some formal language. Thus, in the language of metamath, a variable is any greek and roman letter that appears in a string of that language.
Apr
18
asked Is there a way of defining the notion of a variable mathematically?
Apr
18
revised The topology of distributions
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