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Apr
7
comment Tensor product in multilinear algebra
@tom, I think what you are asking about is the map $X^*\otimes Y^*\to \Hom(X,Y^*)$, which is not an isomorphism unless $X$ is finite-dimensional... or some other constraint. A different set-up that makes your "Functions" riff exactly true is free objects on sets. But, in general, there'll be obstructions, I think.
Apr
6
comment Tensor product in multilinear algebra
@tom ... good! :)
Apr
6
comment Tensor product in multilinear algebra
@tom, ... let me try again! :) The roles of $\beta$ and $b$ got reversed... I edited again. There's a fixed $b:V\times W\to V\otimes W$, so that for all $\beta:V\times W\to X$ there's $B:V\otimes W\to X$ such that $\beta=B\circ b$. Sorry for my haste!
Apr
6
revised Tensor product in multilinear algebra
added 2 characters in body
Apr
6
comment Tensor product in multilinear algebra
@tom, oops, unfortunate typo: should have been "compose", not "tensor". I edited it. Also, it's not really a map on $V\oplus W$, but on $V\times W$, the cartesian product, not direct sum.
Apr
6
revised Tensor product in multilinear algebra
added 1 character in body
Apr
6
comment Use of $L^2$ norm in calculus of variations
The minimum principle is true in Hilbert spaces, but not in Banach spaces, and the latter failures are not at all pathological. Thus, the original formulation of the "Dirichlet Principle" was not correct, waiting for Beppo Levi's Hilbert-space treatment of it c. 1906.
Mar
25
answered if we change the norm, is it possible to make it complete ?
Mar
25
revised $Aut(D_4)$ is isomorphic to $D_4$?
edited tags
Mar
22
comment Fractional Linear Transformation: Region between two circles to strip
Three points determine a circle and/or line, and linear fractional transformations preserve circles-and-lines. Linear fractional transformations can map any triple of distinct points on the Riemann sphere to any other...
Mar
17
awarded  Nice Answer
Mar
16
comment Intermediate subfields of $k(x)/k$
This question should be re-opened: as the questioner observes, it is not so easy to prove that the obvious-appearing things are true, I think. Sure, it is possible (my algebra notes do such things), but in my experience the supposed obviousness is dismissed incompetently in too many sources... E.g., computation of the Galois group of $k(x)$ over $k(x^n)$ (with $n$th roots of unity in $k$)? Sure, it is do-able, but ... something is required.
Mar
2
revised Finding the biggest decrease in a list of integers (Python)
edited tags
Feb
28
answered Undergraduate Research Thesis
Feb
26
comment Linear functionals which share the properties of the integral
I disagree, but perhaps in a way irrelevant to most of the questioner's purposes/needs.
Feb
26
comment Linear functionals which share the properties of the integral
Tastes vary, but to my mind "monotonicity on a lattice [sic]" is not very explanatory. As in "I'd encourage my competitors to take that viewpoint"... Naturally occurring spaces of functions do have topologies, and integrals and other functionals and linear maps are continuous... "Functional analysis" can connote many things, and be inflated to something unreasonable, but I do honestly think that many such questions are only sensibly answerable in a context of topologies on spaces of functions. I know, people have other opinions...
Feb
26
comment Linear functionals which share the properties of the integral
Continuity of functionals really does matter... and "measure theory" does not come close to explaining what the topology might be on the space of continuous, compactly-supported (real-valued, or complex-valued) functions on the real line... It would be misleading to suggest to people that answers to such questions are to be found in the usual measure-theory texts, or "real analysis" texts... unfortunately.
Feb
26
comment Linear functionals which share the properties of the integral
[continuation] not usually covered "even" in functional analysis courses, which usually do not substantially develop these ideas.
Feb
26
comment Linear functionals which share the properties of the integral
To be precise, it is the Riesz-Markov-Kakutani theorem (bad communication among different groups at the time!) that asserts that any "(positive? continuous... what topology?) functional on the LF space of compactly-supported continuous functions is given by a measure. Yes, this is usually attempted with a grossly inadequate set-up... The translation-invariance, both existence and uniqueness, are a somewhat different point. The existence is not hard in concrete cases. Uniqueness of "invariant functionals" (especially on test functions...) is non-trivial, and ... [contd]
Feb
20
comment What is the quotient $\mathbb{T}^3/\mathbb{Z}_2$?
And, yet-again, $\mathbb Z_2$ will be read as "$2$-adic integers" by most mathematicians...