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comment Do monomials form a basis for the vector space of real analytic functions?
@user1952009 What is the most natural extension of a concept, if not literally the same concept?
May
1
awarded  general-topology
Apr
30
comment Torus with a point deleted is not a retract of the torus.
@JohnHughes A compact set can be a retract of a noncompact set, but not conversely, since continuity preserves compactness (and the question is about the converse situation). But you are right, the remained of the proof applies to the torus minus an open disk : ).
Apr
30
comment Torus with a point deleted is not a retract of the torus.
@Mark One way to show this is using the fact that $T$ is Hausdorff and connected. It $T \backslash \{x_0\}$ were compact, it would be closed. But it is open, since it is the complement of a point. Since $T$ is connected, then $T \backslash \{x_0\}$ must not be closed. You can also argue directly that it will not be closed, since $x_0$ is clearly a limit point of its complement. Another way is that the torus is a compact metric space, hence complete. If you remove one point, it is not complete anymore (since such a point was clearly not isolated). Therefore, it cannot be compact.
Apr
30
revised Torus with a point deleted is not a retract of the torus.
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Apr
30
revised Torus with a point deleted is not a retract of the torus.
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Apr
30
answered Torus with a point deleted is not a retract of the torus.
Apr
30
comment Compact-open topology on $\operatorname{Hom}_\mathbb{R}(V,W)$
Sorry, I don't understand what you mean by "the topology is given by the vector space structure". What I mean is that, given any two topological vector spaces of the same dimension $X,Y$ (that is, the topology on $X$ and $Y$ are only assumed to be Hausdorff and to make the sum and multiplication by scalar continuous), then any isomorphism between them is a homeomorphism. If you have two different topologies on a vector space $X$, if both of them make $X$ a TVS, then the identity will therefore be a homeomorphism. This fact is a consequence of Theorem $1.21$ of Rudin's FA.
Apr
30
comment Compact-open topology on $\operatorname{Hom}_\mathbb{R}(V,W)$
In the case of finite dimension, yes. In finite dimension topological vector spaces, every isomorphism is a homeomorphism. The identity in a vector space is a isomorphism, hence you have your result. If one or both are not finite dimensional, first of all we don't have $\text{Hom}_{\mathbb{R}} (V,W) \subset C^0(V,W)$. Secondly, you must tell what topologies you are considering. However, I don't know any general result.
Apr
29
revised How to show that a continous function $f:\mathbb{R}^m \to \mathbb{R}$ has a maximum?
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Apr
29
answered How to show that a continous function $f:\mathbb{R}^m \to \mathbb{R}$ has a maximum?
Apr
29
answered Differentiability of vector valued function?
Apr
27
answered Argue that the iterated integral of a continuous function is continuous
Apr
26
comment $T:R^3 \rightarrow R^3$. Show that there is a line L such that $T(L) =L$.
What is the problem?
Apr
25
comment Prove $\sum_{n = 0}^{\infty} a_n X^n$ converges at every point in $[-1,1)$ if $a_n$ is non-increasing and converges to $0$
Do you mean $\sum a_n X^n$? Otherwise, what is the radius of convergence of a series of real numbers?
Apr
24
revised Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.
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Apr
24
answered Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.
Apr
24
revised Range of this double trigonometric function
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Apr
24
answered Why $\displaystyle\sum_{n=1}^\infty {\ln(n^2) \over (n^2) }$ converges?
Apr
24
revised Why does dual basis not span the dual space when the given vector space V is infinite dimensional?
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