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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27 views

Interchanging Limits: $a_{n,k}\to C$ implies $\sum_k^n a_{n,k}b_k\to C\sum_k^\infty b_k$?

Consider a doubly-indexed sequence $(a_{n,k})$ which, for each $k$, converges as $n\to\infty$ to an absolute constant $C$. In what generality can we say that for a sequence $b_k$ ...
4
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1answer
83 views

Universal property of topology of uniform convergence

What kind of universal property does the strong dual topology on $X'$ have, for $X$ being a locally convex space. Is it possible to define $X'$ as the projective limit of the normed spaces ...
2
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1answer
40 views

For $W'\subset W$ with $W\in B,W'\in B'$ bases of topologies $T,T'$; we get $T\subset T'$

Let $A$ be a set with $T,T'$ topologies, and $B,B'$ bases for $T, T'$ respectively. 1 Suppose for all $a\in A$ and $W\in B$ with $a\in W$ there exists a $W'\in B'$ with $a\in W'$ and $W'\subset ...
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0answers
13 views

A question on Cauchy filters

Let $\mathfrak{F}$, $\mathfrak{G}$ and $\mathfrak{G_1}$ be Cauchy filters in a uniform vector space $(X, \mathfrak{U})$. Let $c\in \mathbb{C}$. (1) If $\lim (\mathfrak{G}- \mathfrak{G}_1) = 0$, can ...
3
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1answer
213 views

Continuous inclusions in locally convex spaces

Let $(X, \left \| \cdot \right \|_X )$, $(X, \left \| \cdot \right \|_Y)$ two normed vector spaces with $X \subset Y$, by definition we have $X \hookrightarrow Y$ if $\left \| x \right \|_Y \leq C ...
3
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0answers
49 views

What dimensions permit a cross product [duplicate]

A cross product is possible in a $3$D and in a $7$D system. What prevents a cross product from being possible in a vector system of higher number of dimensions? For instance $15$D or $2^n$$-1$ ...
3
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37 views

When is a topology is defined for this special kind of uniform convergence?

Question: Let $\phi:X\to X$ be bijective and continuous. Does a topology $\tau$ exist such that $$f_n\overset\tau\to f\Leftrightarrow \phi\circ f_n\overset\infty\to\phi\circ f$$ where $\infty$ ...
0
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1answer
28 views

Show that $\{(x,y) \in \mathbb{R}^2 : |x|+|y|=2\} \subset \partial A$

Let $A= \{(x,y) \in \mathbb{R}^2 : |x|+|y|<2\}$. Show that $\{(x,y) \in \mathbb{R}^2 : |x|+|y|=2\} \subset \partial A$. As all norm are equivalents in $\mathbb{R}^2$, it is reasonable the use ...
0
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0answers
20 views

Distributions on compact and semi-open intervals

In the theory of distributions (aka generalized functions), one considers mostly distributions $T \in \mathcal{D}(\Omega)$ on an open subset $\Omega \subseteq \mathbb{R}^n$. Hereby, the space ...
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0answers
33 views

Is a strongly holomorphic function automatically continuous?

I haven't found any sources that state this explicitly for arbitrary topological vector spaces (most sources are concerned with Frechet spaces, where even weakly holomorphic implies continuous). I'm ...
3
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1answer
38 views

Infinite dimensional topological vectorspaces with dense finite dimensional subspaces

Consider $\mathbb R$ as a $\mathbb Q$ vector space. Using the usual metric on $\mathbb R$, we find: $\mathbb Q \subset \mathbb R$ is dense and one dimensional (indeed every non-zero subspace appears ...
1
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1answer
36 views

Closed subsets of $\mathbb{C}^*$ proper for multiplication

Let $S_1$ and $S_2$ be two proper closed subsets of $\mathbb{C}^*$. Let's denote by $\overline{S_1}$ and $\overline{S_2}$ their closure in $\mathbb{C}_{\infty}.$ (Alexandrov compactification) ...
0
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0answers
19 views

Dense convex set in $*$-weak topology

Let $X$ be a Hausdorff topological vector space over $\mathbb{K}$. Suppose $W$ is a convex subset of its topological dual $X'$. How to prove that if for any $x\in X\setminus\{0\}$ set $\{f(x):f\in ...
1
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1answer
19 views

Bounded neighbourhood of zero in TVS

Is is true that in any topological vector space, which is $T_1$ there exists bounded neighbourhood of zero ? Is is still true if we omit $T_1$ axiom ?
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0answers
17 views

Space of polynomials as a continuous image of F-space

Let $X=\mathbb{R}[a,b]$. Is there any norm $\|\cdot\|$ on $X$ s.th. $X$ is a continuous image of some $F$-space. ($F$-space means that there exists complete metric s.th. $d(x+z,y+z)=d(x,y)$) ? My ...
9
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1answer
230 views

Why do we give $C_c^\infty(\mathbb{R}^d)$ the topology induced by all good seminorms?

Briefly, my question boils down to the following: What benefits do we gain from considering the space of test functions in the topology induced by all "good" seminorms, as opposed to other topologies ...
0
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0answers
75 views

When is a diffeomorphism analytic?

I've read somewhere "a class $C^\infty$ diffeomorphism is said to be analytic" but I forgot to write down where I read this and now I can not find it, which makes me wonder if it's true? I'm working ...
0
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0answers
50 views

Looking for an example of a bounded set.

Consider the local base over the space of complex continuous functions over $[0,1]$ (denoted by $\mathcal{C}[0,1]$) defined for each fixed $x\in [0,1]$ and $\epsilon>0$ by ...
1
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1answer
64 views

6.5 theorem in Functional Analysis by Rudin about topological vector space and $\mathscr{D(\Omega)}$

I am reading the proof of 6.5 Theorem in Rudin's book. For part (c), he wants to prove that a bounded subset $E \subset \Omega$ must be contained in a $\mathscr{D}_{k}$ for some compact subset $K ...
0
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1answer
54 views

Is a sequentially compact (non-metrizable) uniform space totally bounded?

First some topological definitions in terms of nets and sequences: A topological space $(X, \tau)$ is compact iff every net has a convergent subnet sequentially compact iff every sequence has a ...
1
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1answer
38 views

Open sets in the unitary group $ U(\mathcal{H}) $ of a Hilbert space $ \mathcal{H} $.

Let $H$ be an infinite dimensional Hilbert space and let $(x_i)_1^\infty$ be an orthonormal basis for $H$. Consider $U(H)$ the unitary group of the continuous unitary operators on $H$. Equip $U(H)$ ...
0
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0answers
38 views

embeddings of TVS are continuous

How to show that the two embeddings of TVS are continuous: $C^{\infty}_c\subset S\subset C^{\infty}$? According to Wikipedia, the definition of "Continuously embedded" is that " one normed vector ...
0
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0answers
32 views

Check proof that the embedding of the unit ball $B\subset X$ into $X^{**}$ in weak-* dense

I have to prove the following theorem: Let $X$ be a (real) Banach space, and let $B$ denote its closed unit ball, and let $\tau (B)$ denote its canonical embedding into $B^{**}$, the closed unit ...
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5answers
644 views

Nonconstant linear functional on a topological vector space is an open mapping

In the middle of another proof (Theorem 3.4, p. 60) in his Functional Analysis book, Rudin says that "every nonconstant linear functional on $X$ (topological vector space) is an open mapping." Is ...
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0answers
43 views

Existence of linear continuous functional on locally convex TVS

Let $c>1$ and $X$ be a locally convex TVS. Take $a,b\in X$ and a closed subspace $Y$ of $X$ such that $Y\cap \{(1-t)a+bt \ | \ t\in\mathbb{R}\}=\emptyset$. How to prove that there exist $f\in X'$ ...
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1answer
120 views

$V$ is finite dimensional iff $V'$ with the weak topology is normable

Why is the following statement valid? Note, $V$ is locally convex Hausdorff topological vector space over $\mathbb{C}$ and $V'$ is the space of all continuous linear maps from $V \to \mathbb{C}$. ...
1
vote
1answer
45 views

Locally bounded topological vector spaces

Let ‎$ X $ ‎‎‎be a topological vector space ‎such that every neighborhood of zero contains an infinite dimensional subspace. Then ‎$ X $ is not locally bounded. I do not know, why? We know ‎$ X $ is ...
0
votes
2answers
76 views

Is Topological Space a Metric Space?

What's the correct relationship between these two spaces? I think that topological space is a metric space, since the open is defined by a metric such that $d(x, a) < \epsilon$.
3
votes
1answer
42 views

The weak topology on $H$ is the weak* topology on $H^*$ pulled back via $\Phi$

I'm reading the following in Analysis Now by Pedersen: The map, $H$ a Hilbert space $$\Phi:H\to H^*: x\mapsto(\cdot\mid x)=[y\mapsto (y,x)]$$ is a conjugate linear isometry. Then define the weak ...
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1answer
21 views

Does $\frac{1}{n}x_0$ converge to the origin in any topological vector-space?

Let $X$ be a topological $\mathbf{R}$-vector-space (not necessarily Hausdorff) and $x_0 \neq 0$ a non-zero element of $X$. Then intuitively the sequence $(\frac{1}{n} x_0)_{n \in \mathbf{N}}$ ...
1
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1answer
51 views

Complete as a semimetric space but not as a topological group

I shall begin with some definitions. 1) Suppose that $X$ is a topological (additive) group and $(x_{s})\subseteq X$ is a net, we said that $(x_{s})$ is Cauchy whenever $U$ is a neighbourhood of $0$, ...
0
votes
1answer
46 views

Proof of a property regarding weak-star topology

I am reading a book in which a theorem is only proven analogously in the situation of weak topology. The proof does not seem to work in the case of weak$^{*}$ topology. Let $X$ be a normed linear ...
2
votes
2answers
24 views

If $V$ is completely normable, then is every norm complete?

Here is a theorem that motivated my question. Let $(V,||\cdot||_V)$ be a normed space over $\mathbb{K}$. Then, there exists a Banach space $(X,||\cdot||_X)$ such that $V$ is dense in $X$ and ...
0
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0answers
11 views

Is there a terminology to denote a topological vector space that admits an inner product?

Let $(V,\tau)$ be a topological vector space over $\mathbb{K}$. If there is a norm $||\cdot||$ on $V$ such that the metric topology induced by $||\cdot||$ is $\tau$, then we call $V$ is normable. ...
4
votes
1answer
91 views

Example of a subnet that have no subsequence.

I have an elementary question on nets because I'm not familiar with this concept. Here are two basic facts: Every subsequence of a sequence is a subnet; Not every subnet of a sequence is a ...
0
votes
1answer
37 views

open and closed unit balls in topological vector space

It is a famous fact that in any seminormed space $X$, the open unit ball $B$ and closed unit ball $C$, which are defined by $B=\{x\in X: \|x\|_{X}<1\}$ and $C=\{x\in X: \|x\|_{X}\leq 1\}$ ...
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0answers
22 views

Core points of a convex set

In the book of Gamelin "Unifrom Algebras" I found the following definition: Let $K \subset V$ be a convex set in a vector space. An element $x \in K$ is called core point of $K$ if whenever $z ...
3
votes
2answers
60 views

Example of a topological vector space which is not locally convex

I'm currently studying Functional Analysis and the professor gave an example for a TVS (which we have defined to be a vector-space $X$ in which addition $X \times X \rightarrow X, (x, y) \mapsto x + ...
2
votes
1answer
49 views

Quotient map $\pi : X \rightarrow X / \mathrm{ker}(A)$ is open for a bounded linear operator $A$

I'd like to show: if $A : X \rightarrow Y$ is a bounded linear operator between Banach-spaces, then $\pi : X \rightarrow X / \mathrm{ker}(A)$ is a open map. I found a proof, which I do not really ...
1
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1answer
48 views

Dual topology and Mackey–Arens theorem

I read only by wikipedia the Mackey–Arens theorem, that is: Given dual pair $(X, X')$ with $X$ a locally convex space and $X'$ its continuous dual, then $\mathcal{T}$ is a dual topology on $X$ if ...
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1answer
13 views

Semi-norm generating the standard locally convex topology on the space of locally finite Borel measures

In several articles available over the internet, it is written that: .. $M_{loc}(\Omega)$ is the space of locally finite Borel Measures on $\Omega$ with the standard locally convex topology ...
4
votes
2answers
50 views

A $T_0$ topological vector space is Hausdorff

I know a $T_0$ topological group is $T_1$, and if we have a topological vector space, there should be a way of using scalar multiplication to get disjoint neighborhoods. Right?
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0answers
79 views

Local boundedness of monotone operators in general spaces

A classical result states that: If $X$ is a Banach space then every multi-valued monotone operator $T:X\to 2^{X^*}$ is locally bounded on $\operatorname*{int}D(T)$ (the interior of its domain). I ...
2
votes
1answer
58 views

In the Krein-Milman theorem, can the weak closure of the convex hull be replaced by norm-closure?

I have a question on the following formulation of the Krein-Milman theorem: Consider a vector space $X$ equipped with the weak topology induced by a separating space $X^*$ of functionals on $X$. ...
0
votes
2answers
70 views

Continuity of seminorms

The following is from Wikipedia: A locally convex space is defined to be a vector space $V$ along with a family of seminorms $\{p_α\}_{α ∈ A}$ on $V$. A locally convex space carries a natural ...
1
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1answer
82 views

Equivalent definitions of locally convex topological vector space

This Wikipedia article gives two equivalent definitions of locally convex space (l.c.s). I don't see clearly the equivalence and I'd like to make it crystal clear. Definition 1 Let $(V,\tau)$ be a ...
2
votes
1answer
43 views

Different definitions of absorbing sets from the Wikepedia

Consider a vector space $X$ over the field $\mathbb{F}$ of real or complex numbers and a set $S\subset X$. In this Wikipedia article about absorbing sets, $S$ is called absorbing if for all ...
4
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1answer
92 views

Discontinuous surjective linear map which is not open

The following statement is true: Assume that $X$ and $Y$ are topological vector spaces where $Y$ is finite-dimensional Hausdorff, if $A:X\rightarrow Y$ is a continuous surjective linear map then $A$ ...
6
votes
2answers
149 views

Proving that scalar multiplication is continuous

Let $\mathbb{K} \in \{ \mathbb{R} , \mathbb{C} \}$ and $s= \mathbb{K}^{\omega}$ be the usual sequence set with entries on $\mathbb{K}$. I proved that $\mathbb{K}$ induces a $\mathbb{K}$-vector space ...
0
votes
4answers
96 views

Are the axioms of a topological space superfluous?

A topology on a set $X$ is a family $\mathcal{T}$ of subsets of $X$, which are open sets and satisfy: (1) $\emptyset, X \in \mathcal{T}$. (2) Any union of elements of $\mathcal{T}$ belongs to ...