# Tagged Questions

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.

179 views

### Nuclear spaces vs Banach spaces

The Wikipedia article on nuclear spaces say the following: "There are no Banach spaces that are nuclear, except for the finite-dimensional ones. In practice a sort of converse to this is often true: ...
65 views

### Closed Monoidal Structures On The Category Of Complete Topological Vector Spaces

Context: The category of Banach spaces, with the projective tensor product is a closed monoidal category. Question 1: Is there a tensor product on the category of complete topological vector spaces, ...
168 views

### Isomorphism between spaces of sections.

Let $M$ be a compact manifold and let $E_i \xrightarrow{\Large \pi_i} M$, $i = 1, 2$, be two (real or complex) vector bundles of the same rank $k$ over $M$. Assume we have metrics $g_1, g_2$ on $E_1$, ...
142 views

### Connections and dependences between topological and algebraic basis in topological vector space

On my last functional analysis exam, one of the tasks was to show that if normed vector space $X$ have countable Hamel basis, then $X$ is separable space (over field $\mathbb{R}$). I am not sure if ...
87 views

### Infinite Dimensional Topological Vector Space

Let $V$ is a finite-dimensional vector space over $\mathbb{R}$ (or $\mathbb{C}$). To make $V$ a topological space, we may choose the sets $f^{-1}(U)$ as a sub-basis, where $f$ ranges over all linear ...
152 views

### Curiosities about the content of a rare book: Topological Vector Spaces by A. Grothendieck

The book is a celebrated and highly influential book by A. Grothendeck, which was published in 1954, in French and for various reasons, it has been out of print since 1973. I am very much interested ...
39 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 $\infty$ ...
95 views

### Are uncountable “Schauder-like” bases studied/used?

We could define the following notion of basis in a way analogous to unconditional Schauder basis: If $X$ is a topological vector space over $\mathbb R$ and $B=\{b_i; i\in I\}$ be a subset of $X$. ...
133 views

### The topology of $GL(V)$

Let $V$ be a topological vector space (not necessarily finite-dimensional) over a field $K$, and let $GL(V)$ be the group of invertible linear maps $V\to V$ under composition. There are two obvious ...
73 views

### Schwartz space of functions versus Schwartz space in a more general sense?

Part of me is afraid that this isn't a well-formed question, but try as I might, I can't seem to figure out anything reasonable on this topic. I'm hoping someone here can help. In functional analysis,...
51 views

### How do they call the topological tensor product that classifies operators from Hilbert space?

Let $V$ and $W$ be topological vector spaces. There are different ways to complete the tensor product $V \otimes W$, and the only ones that are usually discussed in introductory literature are the ...
252 views

### The converse of James's Theorem

The famous James theorem states that: Theorem. Let $X$ be a (Hausdorff separated) locally convex space (LCS for short) with topological dual $X^*$ and let $B\subset X$ be weakly-closed. If $X$ is ...
50 views

### Extending the topology on a set to the group it generates

The multiplicative group $\Bbb Q^+$ can be viewed as a $\Bbb Z$-module. To see this, note that any rational can be decomposed into the form $2^{n_2} \cdot 3^{n_3} \cdot 5^{n_5} \cdot ...$ The tuple ...
61 views

### Reference request for Grothendieck's work on “Integration with values in a topological group”

Recently I was reading the available part of the second part of W. Scharlau's book on Alexandre Grothendieck (see here). There I found, An anecdote survives about Grothendieck's arrival in Nancy: ...
80 views

### When weakly compactness implies compactness?

Let $A$ be a Banach space. The weak topology on $A$ is a topology which produced by the following family of seminorms: $~~~~~~~~~~~~~~~~~~~~P_f(x)=|f(x)|,\qquad$ where $f\in A^*$ and $A^*$ is dual ...
69 views

### Closedness of convex sets in a locally convex space

Let $C$ be a convex subset of a locally convex topological vector space. Consider the properties: a) $C$ is closed. b) $C$ is weakly closed. c) $C$ is weakly sequentially closed. d) $C$ is ...
143 views

### What do we call a Schauder-like basis that is uncountable?

In a topological vector space, every Schauder basis is assumed countable, by definition. Supposing we drop the countability condition, we call this a [what goes here?] basis?
69 views

### How general is the convergence of the exponential function's power series?

Let $\mathbf{V}$ be a Fréchet space whose underlying set is $V$. Let $\;\; \beta \: : \: V\times V \: \to \: V \;\;$ be a continuous bilinear map that has an identity element and is power-...
68 views

### Every locally connected non separating plane continuum is an AR

A continuum is a compact connected metric space. A plane continuum is a continuum contained in $R^2$. We say that a plane continuum does not separate the plane provided that $R^2\setminus X$ is ...
83 views

### Pseudo norm-exercice

Let $f$ be a measurable function with finite values almost everywhere. We put $$N_0(f) = \displaystyle\int \dfrac{|f|}{1 + |f|} d \mu.$$ We denoted by $L^0$ the set of measurable functions $f$ such ...
144 views

### Which (endo)functors of the category of finite-dimensional real vector spaces induce continuous maps between Hom-sets?

Let $\operatorname{Vect-fin}$ be a category of finite-dimensional vector spaces over $\mathbb{R}$. In this category Hom-sets $\operatorname{Hom}(V,W)$ are themselves finite-dimensional vector spaces ...
111 views

### Density of operators

I am interested in operators on non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{**})$ by $\Phi(T)=T^{**}$;...
719 views

### Translation invariant metric

Under what conditions can a metric vector space be given an equivalent metric that is translation invariant? I was wondering if the probability measures on real line can be embedded in vector space ...
345 views

113 views

27 views

76 views

232 views

### Continuity of implicit function

I find a proof of the following theorem in A.N. Kolmogorov and S.V. Fomin's Элементы теории функций и функционального анализа (pp. 492-493 here): Let $X,Y,Z$ be Banach spaces, $U$ a neighbourhood ...
63 views

### Limit topology of a sequence of topological vector spaces

Under which circumstances is the limit topology of an increasing sequence $E_0\subseteq E_1\subseteq E_2\subseteq\cdots$ of topological vector spaces, where the inclusion maps are linear and ...
57 views

### Topological vector space question

$C[0,1]=$ space of all continuous complex valued function over $[0,1]$. Define metric, $d(f,g)={\int_{0}^{1} \frac {|f(x)-g(x)|}{1+|f(x)-g(x)|}}$, for all $f,g\in C[0,1] .$ Let $(C[0,1],\sigma)$ ...
47 views

### Metric space with measure and a special property

Let $R$ be a metric space endowed with a (complete) measure $\mu$ satisfying the following condition: all the open and closed sets of $R$ are measurable and for any measurable set $M\subset R$ and any ...
43 views

### Extension of function with values in a Banach space

I want to prove the following Let $E,X$ be Banach spaces, and $Y\subset E$ a closed subspace with codimension $1$. Let $T:Y \to X$ be a continuous linear function. Then there exists a continuous ...
25 views

### T1 axiom in dual space

My books talks about the conjugated space, but does it mean dual space? Are not the same thing? I don't understand why in the dual space $E^\ast$ of $E$, the separation axiom $T_1$ is satisfied and ...
28 views

### When a sort of weak topology is enough to generate vector space topology

Consider a vector space $V$, and some functions $f_\alpha: V \rightarrow \mathbb{C}$ where $\alpha$ ranges over some index set $A$. We can think about the coarsest topology which: a) makes the \$f_\...