# 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.

21 views

### Let A⊆B, If B has second category itself then A has second category itself

Let $X$ be topological vector space and $A⊆B$. If $B$ has second category itself then $A$ has second category itself. Is this true?( $A$ is not empty)
65 views

26 views

### Isomorphism of finite dimensional topological vector space with $(\mathbf{R}^k,\mathcal{R})$

Let $(T,\mathcal{T})$ be a topological vector space over $\mathbf{R}$ with finite positive dimension. Is it true that there exists an isomorphism between $(T,\mathcal{T})$ and ...
30 views

### Generate a mesh from unsorted points (eight points)

I'm trying to generate a mesh from eight points. The challenge is that I don't know the order/label of the points, and I want it to work regardless of variations in the shape (see example below). The ...
50 views

### Weak* topology on Hilbert space

I am a little confused about the weak* topology on Hilbert space $H$. Beyond doubt, the weak* topology on $H^{**}$ is $\sigma(H^{**},H^*)$. Suppose $\tau$ is the natural embedding from $H$ onto ...
128 views

### Embedding vs continuous injection (in topological vector spaces)

When working with topological vector spaces (say $X,Y$), the term “embedding” is often used for a continuous injection $f:X\rightarrow Y$. Now, $f$ is of course a bijection onto its image, but it's ...
206 views

### Generalization of inner product spaces (analogue to uniform spaces/locally convex spaces)

In the following I am going to devise a chart of topological spaces that contains inner product spaces, normed vector spaces, metric spaces and other related spaces. In the end there will be a gap in ...
71 views

### Open neighbourhoods in topological vector spaces

It is well known that each open ball in a Banach space is homeomorphic to the whole space. Can we extend this to topological vector spaces? In other words, does every non-void open set in a ...
21 views

### A question involving normed spaces and strictly convex spaces

Let $(X, \| \cdot \|_X)$ be a normed space and let $\| \cdot \|$ be a norm on $X$ such that $(X, \| \cdot \|)$ is strictly convex. How can I find a strictly convex space $(Y, \| \cdot \|_Y)$ and a ...
152 views

### Is any closed and bounded subset of a reflexive Banach space compact in the weak topology?

It seems to me that Alaoglu's theorem implies that any closed and bounded subset of a reflexive Banach space is compact in the weak topology. Is convexity of the set also needed?
55 views

### Why doesn't Alaoglu's theorem imply that $X^{*}$ is locally compact in the weak* topology?

I must be missing something basic and simple: If $X$ is a normed vector space and the closed unit ball in $X^{*}$ is weak* compact, and translations and dilations are homeomorphisms, why isn't $X^{*}$ ...
32 views

### If $E\subset X^{*}$ is bounded, then so is its weak* closure

If $X$ is a Banach space and $E\subset X^{*}$ is norm-bounded, I've shown that its weak* closure is also norm-bounded using Alaoglu's theorem. But perhaps using Alaoglu's theorem is not necessary? ...
30 views

133 views

### Topology generated by a Family of Seminorms as a Initial Topology?

Let $X$ be a set and $\{(Y_i, \mathscr{T}_i)\}_{i\in I}$ be a family of topological spaces and $\{f_i\}_{i\in I}$ a family of mappings $$f_i:X\longrightarrow Y_i.$$ The initial topology on $X$ is the ...
36 views

### Bornological/Barrelled Operator-Topologies?

I'm looking for results concerning the following questions. If those have been already addressed in the literature, it would be nice to know proper citations: Let $(E, \tau_E)$ and $(F, \tau_F)$ be ...
221 views

### Is the dual space of all Radon measures the space of signed measures on a $\delta$-ring?

Consider the Banach space $C_c(\mathbb{R})$ of continuous functions with compact support equipped with the uniform norm $||f||_\infty := \sup_{x \in \mathbb{R}} |f(x)|$. Then it is known (Riesz ...
141 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: ...
45 views

### Continuity of the dual product reloaded

Let $X$ be a Banach space with topological dual $X^*$. Then the dual product $(x,x^*)\in X\times X^*\to x^*(x)\in\mathbb{R}$ is strongly$\times$strongly continuous in $X\times X^*$. That does not ...
### How to prove $\mathit {X}$ is path connected?
${\mathbb{R}}^{2}$ Euclidean 2-space,Let $\mathit {X} \subset \mathbb{R}^{2}$.$$\mathit {X}=[-2,2]\times[-1,0]\cup[-2,-1]\times[0,1]\cup[1,2]\times[0,1]$$ $\qquad\qquad\qquad\qquad\qquad$ ...
In the book A Course in Functional Analysis by Conway, there is the following problem: Problem. Let $X$ be a completely regular topological space, and let $C(X)$ denote the set of all ...