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Results tagged with topological-vector-spaces
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user 61536
A topological vector space is a vector space $V$ over a topological field $\mathbb{K}$ (typically $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$), together with a topology on $V$ such that vector addition and scalar multiplication are both continuous. Hilbert spaces and Banach spaces are examples of topological vector spaces.
2
votes
Inductive limit of $\mathbb R^n$s is Hausdorff and second countable?
Among (infinite-dimensional) topologists the projective and inductive limits of the Euclidean spaces are denoted by $\mathbb R^\omega$ and $\mathbb R^\infty$, respectively. The topology of the (metriz …
- 41.8k
5
votes
Accepted
Is the projectivization of a topological vector space Tychonoff?
The projective space $PE$ of a topological vector space $E$ is Hausdorff but in general is not Tychonoff, not functionally Hausdorff and even not Urysohn (let us recall that a topological space is Ury …
- 41.8k
3
votes
Accepted
Is a topology sandwiched between two norms compactly generated?
Let $\tau$ be the weak topology on the Banach space $\ell_1$. It is known that each weakly convergent sequence in $\ell_1$ is norm convergent (i.e., $\ell_1$ has the Shur property). This property impl …
- 41.8k
2
votes
3
answers
228
views
Every linear topological space embeds into the Tychonoff product of linear metric spaces
I need a reference to the following (known?)
Fact. Every topological vector space $X$ over the field of real numbers is topologically isomorphic to a linear subspace of the Tychonoff product of li …
- 41.8k
3
votes
Is each cometrizable space a subspace of a cometrizable topological group?
At the moment I know the answer to the Question (but not to the Problem). First a definition.
A subset $D$ of a topological space $X$ is called $k$-dense in $X$ if each compact subset $K\subset X$ c …
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3
votes
1
answer
106
views
Is each cometrizable space a subspace of a cometrizable topological group?
Following Gruenhage we call a topological space $X$ cometrizable if $X$ admits a weaker metrizable topology such that every point $x\in X$ has a (not necessarily open) neighborhood base consisting of …
- 41.8k
1
vote
The completeness of spaces of continuous functions with the compact-open topology
This problem has a partial affirmative answer:
Theorem. For any Tychonoff space $X$ the function space $C_k(X)$ is Polish if and only if it admits a stronger Polish locally convex topology.
Proof: T …
- 41.8k
5
votes
1
answer
330
views
Is the compact-open topology on the dual of a separable Frechet space sequential?
Let $X$ be a separable Frechet space (= Polish locally convex linear metric space) and $X'_c$ be the space of linear continuous functionals on $X$, endowed with the compact-open topology (= the topolo …
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2
votes
1
answer
346
views
The completeness of spaces of continuous functions with the compact-open topology
For a Tychonoff space $X$ let $C_k(X)$ denote the space of continuous real-valued functions on $X$, endowed with the compact-open topology.
Problem. Is the space $C_k(X)$ Polish if it is Polishabl …
- 41.8k
4
votes
On convergent sequences in locally convex topological vector spaces
Oh, sorry! I posed this question too quickly.
A simple example is the dual space $\ell_1=c_0^*$ to the Banach space $c_0$, endpowed with the weak$^*$ topology. The sequence $(e^*_n)_{n\in\omega}$ of …
- 41.8k
4
votes
2
answers
433
views
On convergent sequences in locally convex topological vector spaces
Assume that a sequence $(x_n)_{n\in\omega}$ of points of a locally convex topological vector space converges to zero. Is it always possible to find increasing number sequences $(n_k)_{k\in\omega}$ and …
- 41.8k
7
votes
Accepted
Is restriction a closed map?
The answer to the main question is negative:
Consider the compact subset $X=[0,1]\cup \{2\}$ of the real line and let $Y=\{2\}$ be a singleton in $X$. In the function space $C(X)$ consider the closed …
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3
votes
1
answer
200
views
Is each closed subgroups of $\mathbb R^\omega$ isomorphic to a Tychonoff product of locally ...
It is known that any closed linear subspace of $\mathbb R^\omega$ is topologically isomorphic to $\mathbb R^n$ for some $n\in\omega$.
Problem 1. Is each closed subgroup of $\mathbb Z^\omega$ (or bett …
- 41.8k
2
votes
Accepted
Approximation of the identity by finite range functions in topological vector spaces
It seems that the space $X:=C_p(2^\omega)$ of real-valued continuous functions on the Cantor set is a counterexample to this question. The space $C_p(2^\omega)$ is endowed with the topology of pointwi …
- 41.8k
3
votes
Accepted
Measurability of the product on particular topological vector spaces
The answer seems to be "no" even for metrizable separable Banach spaces, which have property $\mathbf P$.
Take any infinite-dimensional separable Banach space $X$, fix a non-zero point $x_0\in X$ an …
- 41.8k