Questions tagged [topological-vector-spaces]
The study of vector space with a topology which makes the maps which sums two vectors and which multiply a vector by a scalar continuous. It's a natural generalization of normed spaces.
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Must a linear functional on a TVS with a sequentially closed kernel be sequentially continuous?
Let $(X, \tau)$ be a Hausdorff topological vector space, $\mathbb{K}$ the scalar field with its usual topology and $\Lambda : X \to \mathbb{K}$ a linear functional.
There is this general criterion ...
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A characterization of closure of a certain class of sets in $\mathbb{R}^n$
Consider a set $K\subset \mathbb{R}^n$ that is symmetric ($B = -B$) and verifies $aK\subset bK$ if $|a|<|b|$. Can I conclude that $\overline{K} = \cap_{a>1}aK$? If not, does the result hold ...
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Theorem 1.36 in Rudin's Functional Analysis
I have a few questions regarding the proof of Theorem 1.36 in Rudin's Functional Analysis:
Why does $V$ being open imply $x/t\in V$ for some $t<1$.
How does the inequality $\mu_V(x-y)<r$ come ...
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Is multiplication by a scalar an open map in topological groups?
If n is any non-zero integer and G is a topological abelian group, under what conditions is the continuous homomorphism $g \mapsto ng$ open or weakly open?
If this map happens to be open for all n and ...
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Proof that every $T_0$ Topological Vector Space is regular
I'm currently reading through Convex Analysis and Beyond by Mordukhovich and Nam where the following preposition and proof are given (note that the authors define TVSs to be $T_0$).
Proposition 1.92 ...
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Is the topology essential for local fields?
References to local fields tend to define local fields in some topological way. For example, a field $K$ is a local field if it is complete with respect to a topology induced by a discrete valuation, ...
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Finding a compact set to apply the Hahn-Banach separation theorem in a locally convex topological vector space
I am trying to justify how the Hahn-Banach theorem was applied in the proof below. It looks like the proof is using the case for locally convex space (because the inequalities are strict). that ...
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Local basis criterion for topological group / vector space?
Let $X$ be a topological vector space (TVS), and let $\mathcal B$ be nonempty family of subsets of $X$ that each contain $0$. Are there simple conditions that guarantee $\mathcal B$ is a local basis ...
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Dual space of a topological vector space that doesn't separate points?
While I am studying FUNCTIONAL ANALYSIS by Walter Rudin, I found the following corollary.
Now, I wonder how the dual space(the set of all continuous linear functionals on $X$) of some pathetic ...
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Can I define boundedness in a topological vector space w.r.t. arbitrary point?
I have started reading Rudin's Functional Analysis, and in Section 1.6, he makes the following definition:
A subset $E$ of a topological vector space is said to be bounded iff for every open ...
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Does the projective tensor product obey a tensor-hom adjunction?
Let $X, Y, Z$ be three lctvs. Then https://ncatlab.org/nlab/show/inductive+tensor+product , the first theorem in section 3, tells us that
$$\operatorname{Hom}(X\otimes_{\iota} Y, Z) = \operatorname{...
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Topology of convergence in measure is not compatible with the vector space structure of measurable functions
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the ...
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Meaning of injective tensor product
Let $X, Y$ be two locally convex topological vector spaces. I can tell myself a story to make $X\otimes_{\pi} Y$ and $X\otimes_{\iota} Y$ (the projective and inductive tensor products, respectively) ...
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Hausdorff separation for the definition of Mackey topology
I am reading the definition of the Mackey topology relative to a dual system $(X,Y)$ and the author (Edwards-Functional Analysis) imposes the condition that the dual system must be separated in $Y$. ...
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Having trouble on understanding some parts of Pedersen's explanation of constructing a locally convex topological vector space.
This is what I have constructed so far.
Let $V$ be a vector space and $\{m_i\}_{i \in I}$ be a separating family of semi-norms on $V$. Define $$f_{i, y}(x) := m_i(x - y)$$We can show that the ...
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Let $E$ be a TVS and $f:E \to \mathbb R^n$ linear. If $\ker f$ is closed, then $f$ is continuous at $0_E$
Previously, I showed that
Theorem Let $E$ be a (not necessarily Hausdorff) real TVS and $f:E \to \mathbb R$ linear. If $\ker f$ is closed, then $f$ is continuous at $0_E$.
The proof is as follows:
...
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If $x_d \to 0_E$ and $\sup_{d\in D} |\lambda_d| <\infty$, then $\lambda_d x_d \to 0_E$
Let $E$ be a (not necessarily Hausdorff) real TVS. I'm trying to solve exercise 13 in these notes by professor Gabriel Nagy
Let $(x_d)_{d\in D}$ be a net in $E$ such that $x_d \to 0_E$. Let $(\...
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Textbook on TVS that contains Theorem 32.2 about the completeness of the space of continuous linear maps
I have come across this thread about TVS. The OP took below screenshot of THEOREM 32.2.
Could you please elaborate on the book from which the screenshot was taken?
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The coordinate map of a one-dimensional TVS is continuous
Let $E$ be a one-dimensional real Hausdorff TVS. We fix $v \in E \setminus \{0_E\}$ and consider the linear map
$$
T: \mathbb R \to E, t \mapsto tv.
$$
To prove that any $n$-dimensional Hausdorff ...
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What is the relationship between convex conjugate and polar set?
Given a duality $\left<X^*,X\right>$ over field $\mathbb R$, and any set $A \subseteq X$, the polar set of $A$ is defined as \begin{align}A^\circ = \{x^* \in X^* | \left<x^*,x\right> \leq ...
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For non second-countable TVS, is the sum of measurable functions again measurable? [duplicate]
Let $(\Omega, \mathcal A)$ be a measurable space and $E$ a topological vector space. Let $f,g:\Omega \to E$ be measurable. I already proved that
Theorem $E$ is second-countable, then $f+g$ is ...
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Closure of balanced convex set and its dilation
While reading a proof, one of the steps was showing that for a (specific) balanced and convex set in a Banach space we had the inclusion $\overline{B} \subseteq 2B$. The proof continued using specific ...
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Why does $C^\infty(\Omega)$ have the Heine-Borel property?
$
\let\uto\rightrightarrows
\let\ii\infty
\let\W\Omega
\let\a\alpha
\let\b\beta
\let\e\varepsilon
\let\d\delta
\let\sbe\subseteq
$
I'm struggling to understand the examples at the end of the first ...
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Why is $C^\infty(\Omega)$ complete?
$
\let\uto\rightrightarrows
\let\ii\infty
\let\W\Omega
\let\a\alpha
\let\b\beta
\let\e\varepsilon
\let\d\delta
\let\sbe\subseteq
$
I'm struggling to understand the examples at the end of the first ...
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If $\overline A$ is compact, then $\overline A=\overline A^w$
Let $X$ be a Hausdorff topological vector space such that $X'$,it topological dual separate points of $X$, and $A\subset X$. I want to know if the following sytatement is true or not?
If $\overline A$ ...
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Scales of a bounded neighborhood of $0$ forms a neighborhood basis of $0$.
Exercise: Let $B$ be an open bounded neighborhood of $0$ in a topological vector space. Show that every neighborhood of $0$ contains a set of the form $\{sB : s\in(0,\infty)\}$.
Hint: One can use the ...
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Equivalent definitions of bounded sets in topological vector spaces
I want to show equivalence of following two definitions:
Definition 1: A subset $U$ of a topological vector space is called bounded, if for every neighborhood of $0$ $V$, there is a scalar $s\in\...
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On a class of compacts broader then Eberlein's
The class of Eberlein compacts (those compacts spaces homeomorphic to a weakly compact subset of a Banach space) is well known and well studied; one of the many properties it enjoys is that every set ...
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Is the Fréchet quotient space given by the induced seminorms?
We admit the following definition of Fréchet space.
The definition of Fréchet space:
Let $X$ be a linear space and let $\{p_i:X\to\mathbb R^+\}_{i\in \mathbb N}$ be a countable family of seminorms ...
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Sum of symplectic complement subspaces closed?
I have the following problem (p.73 in Marsden's 'Introduction to Mechanics and Symmetry' Book):
Given is an infinite-dimensional Banach space $Z$ and on it a weakly non-degenerate, symplectic form $\...
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The Hörmander symbol space $S^{-\infty}(\Omega \times \mathbb{R}^n) \subset S^{m}_{cl}(\Omega \times \mathbb{R}^n)$ is closed
This is Exercise 3.4) in Peter Hintz's Introduction to microlocal analysis
Which I am using for exam preparation.
Let $\Omega \subset \mathbb{R}^n$ open.
Consider for $m \in \mathbb{Z}$ the space $S^m(...
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Functional equation on a domain with empty interior in $R^3$
I am interrested in solving a Pexider equation on a restricted domain with a peculiar structure and I am wondering if one of you could indicate if and how I can solve it.
I was asked in a comment to ...
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In a normed space, is the sum of two open sets (open wrt subspace topology of two complements) open?
Let $(V,\|\cdot\|)$ be a (possibly infinite-dimensional) normed space and $V_1,V_2 \subseteq V$ be subspaces such that $V = V_1 \oplus V_2$. Let $B^i_{r_i} \subseteq V_i$ be open balls in $V_i$, wrt ...
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$x_\beta \rightarrow x$ in a locally convex space if and only if $\rho_\alpha(x_\beta, x) \rightarrow 0$ for every seminorm $\rho_\alpha$
Let $X$ be a locally convex space with the family of seminorms $\{p_\alpha\}$. I am trying to get a feel for convergence of nets (or sequences) in these spaces aside from the general topological ...
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Unit semiball of the supremum of seminorms is closed
I don't know how to prove the second part of this exercise (ex. 7.2 in F. Treves, "Topological vector spaces, distributions and kernels").
Let $\mathcal{P}$ be a family of continuous ...
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Confusion about locally convex (topological vector) spaces
The author in the book I am reading defines a locally convex space as follows.
Definition: A topological vector space $(X,\tau)$ is called a locally convex space if there is an index set $A$ and a ...
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Subbase for initial topology of quotient mappings (locally convex spaces)
Consider a family $\{p_i\}_{i\in I}$ of seminorms defined on $X$ and consider for each $i\in I$ the quotient mapping $q_i:X\rightarrow X/\ker(p_i)$. I know, that each quotient $X/\ker(p_i)$ is a ...
- 2,133
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Degenerate convex function that is not finite on the closure of some convex set on which the function is finite.
Let $E$ be a locally convex TVS, $C$ a convex subset of $E$ and $g:E\to]-\infty,\infty]$ a convex function. I was trying to prove the following
If $g$ is finite on $C$, then $g$ is finite on $\...
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Subbase for initial topology
For an index set $I$, there shall be topological spaces $(Y_i,\tau_i)$ with a subbase $S_i$ and mappings $f_i:X\rightarrow Y_i$ for any $i\in I$. Let $\tau$ be the initial topology with respect to $(...
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Neighborhood of $0$ is absorbent in topological vector spaces
Consider a topological vector space $(X,\tau_X)$ and $N_0$ a neighborhood of $0$. I want to show, that it is absorbent, i.e. $X=\cup_{t> 0} tN_0$.
My idea would be to take an arbitrary $x_0\in X$ ...
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On an infinite dimensional locally convex space, no weakly-continuous semi-norm is a norm
Let $X$ be an infinite dimensional locally convex space that separates points and $X^*$ its dual. I would like to prove that no $\sigma(X, X^*)$-continuous semi-norm is actually a norm.
What I have ...
- 4,253
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Statement on a neighborhood of $0$ in topological vector spaces
Consider a topological vector space $E$ and let $U\subseteq E$ be a neighborhood of $0$. I want to understand the proof of the following statement:
If $K\subseteq E$ is compact and $U$ is open with $K\...
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Topology defined by all neighbourhood filters in topological vector space
In topological vector spaces, translation and dilation are homeomorphism. From this one can proof that for any $x\in X$, where $X$ is the topological vector space, the neighbourhood filter $\mathcal{U}...
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Homeomorphism in topological vector spaces
So on the german wikipedia page of topological vector spaces it is written that, if a topological vector space is a Hausdorff space, then translation by a vector and dilation by a scalar are ...
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Showing that entire analytic functions are dense in Schwartz space
The following theorem is presented as question 15.6 in Treves (1967, 1995). $\mathscr{S}$ is the Schwartz space of functions on $\mathbb{R}^n$, and when one says functions on $\mathbb{C}^n$ are 'dense'...
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A locally convex space has a topology given by a single norm if the topology is generated by finitely many seminorms
This is a problem 1a from section V in Reed & Simon's book on functional analysis. It states:
1a. Prove that a locally convex space has a topology given by a single norm if the topology is ...
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The Mackey topology $\tau(X^*, X)$
Let $X$ be a Banach space and $X^*$ its dual. I would like to better understand the Mackey topology $\tau(X^*, X)$ on $X^*$.
The Mackey topology on $X$, $\tau(X,X^*)$ is defined as the topology of ...
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Is there any way to show that the algebraic tensor product $A[[h]] \otimes A[[h]] \subsetneq (A \otimes A) [[h]]\ $?
Let $A$ be a Hopf algebra over $k.$ Consider the formal power series $A[[h]]$ in $h$ over $A$ endowed with the $h$-adic topology. Then how do we show that $A[[h]] \otimes_{\text {alg}} A[[h]] \...
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Are the weakly bounded subsets of the double dual bounded for the natural topology?
I'm trying to get a better understanding of the topologies on the double dual of a Hausdorff locally convex space (l.s.c.). The following question has then come up, and I'm unable to find an answer.
...
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Inductive limit and denseness
Let $\{E_n, n\in \mathbb{N}\}$ be an increasing sequence of linear subspaces of a vector space $E$, i.e. $E_n \subset E_{n+1}$ for all $n\in \mathbb{N}$ such that
$E = \bigcup_{n\in \mathbb{N}} E_n$. ...
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