# 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.

3answers
575 views

### When do weak and original topology coincide?

Let $X$ be a topological vector space with topology $T$. When is the weak topology on $X$ the same as $T$? Of course we always have $T_{weak} \subset T$ by definition but when is \$T \subset ...
2answers
603 views

### When is a notion of convergence induced by a topology?

I'm interested in sufficient conditions for a notion of sequential convergence to be induced by a topology. More precisely: Let $V$ be a vector space over $\mathbb{C}$ endowed with a notion $\tau$ of ...
3answers
446 views

### Topology on the general linear group of a topological vector space

Let $K$ be a topological field. Let $V$ be a topological vector space over $K$ (if it makes things convenient, you may assume it is finite dimensional). Naive Question: Is there a canonical way of ...
1answer
200 views

### Is the standard structure of a topological vector space on reals unique?

The standard stucture of a topological vector space on reals is this given by the metric d(x,y)=|x-y| on the vector space $\mathbb{R},$ with the field of scalars $\mathbb R$ with standard topology. I ...
0answers
256 views

### Differential forms on fuzzy manifolds

This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me. Question In the usual setting of open subsets of ...
1answer
148 views

### If weak topology and weak* topology on $X^*$ agree, must $X$ be reflexive?

Let $X$ be a Banach space and suppose that the weak topology on $X^*$ agrees with the weak* topology on $X^*$. Must $X$ be reflexive? To prove the contrapositive, it will suffice to assume that $X$ ...
1answer
304 views

### Isomorphisms of Fréchet Spaces

What is the proper notion of an isomorphism between Fréchet spaces? Obviously it should be a linear map. I'm just worried about the analytic structure. Should one be able to order the seminorms on ...
1answer
256 views

### Semi-Norms and the Definition of the Weak Topology

When I was searching for the definition of the weak topology, I found two different definitions. One defines the weak topology in terms of a family of semi-norms, while the other defines it in terms ...
1answer
430 views

### Hahn-Banach theorem: 2 versions

I have a question regarding the Hahn-Banach Theorem. Let the analytical version be defined as: Let $E$ be a vector space, $p: E \rightarrow \mathbb{R}$ be a sublinear function and $F$ be a subspace of ...
2answers
550 views

### If $A$ and $B$ are compact, then so is $A+B$.

This is an exercise in Chapter 1 from Rudin's Functional Analysis. Prove the following: Let $X$ be a topological vector space. If $A$ and $B$ are compact subsets of $X$, so is $A+B$. My guess: ...
3answers
397 views

### Do continuous linear functions between Banach spaces extend?

Just wondering... Let $E$, $G$ be Banach spaces, let $U\subset E$ be a subset of $E$, and let $f:U\rightarrow G$ be a continuous linear function. Can $f$ be extended to a continuous linear function on ...
2answers
694 views

### Dual of a dual cone

Any hint on how to prove the following please: Let $K$ be a convex cone, and $K^*$ its dual cone. Prove that $K^{**}$ is the closure of $K$. Thanks!
1answer
168 views

### Do the notions of weak and weak* convergence coincide for $\ell^1(\mathbb{N})$?

As my friends and I were studying for our real analysis final exam yesterday, we were playing with various examples and found ourselves asking this question: The space $\ell^1(\mathbb{N})$ is the ...
1answer
72 views

### Learning Aid for Basic Theorems of Topological Vector Spaces in Functional Analysis

I am self-teaching myself the basics of functional analysis (e.g. topological vector spaces), and frankly I am starting to get a migraine sorting out/organizing in my head all of the ...
1answer
116 views

### Rotation of $\mathbb{R}^3$ by using quaternion

Express the rotation of $\mathbb{R}^3$ by $\frac{\pi}{3}$ about the $x=y=z$ axis by using quaternions and identifying $\mathbb{R}^3$ with $(i,j,k)$-space. Thoughts: From my point of view, every ...
1answer
218 views

### Contractibility of convex set

Suppose that $\Omega$ is a convex open subset of an infinite dimensional vector space $E$ such that $\Omega$ is not contained in any finite dimensional subspace of $E$. Let $Q_m\subset \Omega$ denote ...
2answers
465 views

### Does every $\mathbb{R},\mathbb{C}$ vector space have a norm?

Is there a canonical way to define on any vector space over $\mathbb{K}=\mathbb{R},\mathbb{C}$ a norm ? (Or, if there isn't, can someone give me an example of a vector space over $\mathbb{K}$ that is ...
3answers
133 views

### Can continuity of inverse be omitted from the definition of topological group?

According to Wikipedia, a topological group $G$ is a group and a topological space such that $$(x,y) \mapsto xy$$ and $$x \mapsto x^{-1}$$ are continuous. The second requirement follows from the ...
1answer
205 views

### Generic topology on a vector space?

For a (possibly infinite-dimensional) vector space $V$, I thought about the following topology $\tau$: Let $O \in \tau$ if every $x \in O$ has the property that for every $v \in V$, there is an ...
1answer
304 views

### The dual of a Fréchet space.

Let $\mathcal{F}$ be a Fréchet space (locally convex, Hausdorff, metrizable, with a family of seminorms ${\|~\|_n}$). I've read that the dual $\mathcal{F}^*$ is never a Fréchet space, unless ...
2answers
68 views

### Alternate definition for boundedness in a TVS

Let $X$ be a topological vector space over $\mathbb R$ or $\mathbb C$. A subset $B\subset X$ is defined to be bounded if for any open neighborhood $N$ of $0$ there is a number $\lambda>0$ ...
0answers
111 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 $E_1$, ...
4answers
101 views

### Books on locally convex topological vector spaces

My friend asked me for a good book about locally convex topological vector space. I'm not familar with this. Could you give me some good references on it?
3answers
145 views

### Is $C([0,1])$ a compact space?

Is $C([0,1])$ (I guesss with the max-norm) a compact space? I have to know that because I want to apply Arzela Ascoli.
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93 views

### Question about Topological Vector Spaces

Let $E$ be a Topological Vector Space and $U$ a bounded set of $E$ with $0\in U$, i.e. given any neighborhood $W$ of the origin, there exist $\alpha>0$ such that $\alpha U\subset W$. Is it true ...
2answers
158 views

### Local base of a topological vector space

I would like to prove that if $B$ is local base for a topological vector space $X$, then every member of $B$ contains the closure of some member of $B$. I would appreciate if somebody can guide me ...
1answer
79 views

### local convexity of $L_p$ spaces

wiki says The spaces $L_p([0, 1])$ for $0 < p < 1$ are equipped with the F-norm they are not locally convex, since the only convex neighborhood of zero is the whole space Why is this so? ...
1answer
242 views

### Sequential and topological duals of test function spaces

Given a test function space, in particular $\mathcal{S}=\mathcal{S}(\mathbb{R}^n)$ (the Schwartz space) or $\mathcal{D}=\mathcal{D}(\mathbb{R}^n)$ (the space of compactly supported smooth test ...
1answer
558 views

### Convex functions and families of affine functions

I know that the supremum of a family of affine functions is convex. Just wondering if it is true (and if so how one proves) that the converse -- any $C^1$ convex function is the supremum of some ...
1answer
88 views

### Pseudonormable Product Spaces

I want to prove that a product $\prod_{i\in I}X_i$ of topological vector spaces is pseudonormable only if a finite number of the factor spaces are also pseudonormable and the rest have the trivial ...
1answer
114 views

### Bounded and compact sets in a subspace of $\mathbb R^{\mathbb N}$

Let $$X= \{u=(u_1, u_2, \ldots): u_n \ne 0 \text{ only for a finite number of terms}\}\subseteq\mathbb R^\mathbb N,$$ with the topology inherited from $\mathbb R^\mathbb N$ (the "pointwise ...
0answers
98 views

### Evaluation map is not continuous always.

Let $E$ be a not normable locally convex space, define $$F: E'\times E\to \mathbb R$$ $$(f,e)\to f(e)$$ I have to show that $F$ is not continuous when $E'\times E$ is given product topology. I was ...
4answers
125 views

### Question on Topological vector space 1

I have numbered this question as (1) because I will be posting series of questions where I don't understand. I hope its allowed. I want to prove the following : If $X$ is a topological vector ...
2answers
156 views

### “The two notions of boundedness coincide for locally convex spaces”

From Wiki The boundedness condition for linear operators on normed spaces can be restated. An operator is bounded if it takes every bounded set to a bounded set, and here is meant the more ...
1answer
1k views

### “Every linear mapping on a finite dimensional space is continuous”

From Wiki Every linear function on a finite-dimensional space is continuous. I was wondering what the domain and codomain of such linear function are? Are they any two topological vector ...
2answers
222 views

### Example of a topological vector space

I have the following question: give an example of a topological vector space $E$ with subspace $M$ and $N$, such that $E = M \oplus N$ algebraically, but not topologically (so $E \ncong M \sqcup N$). ...
2answers
215 views

### If you know the convergent sequences, how do you know the open sets?

I have a homework problem which I feel should be simple but is actually surprisingly tricky. This is why I love math sometimes.... Let $X$ be a normed linear space. Suppose $\|\cdot\|_1$ and ...
2answers
184 views

### Closed Bounded but not compact Subset of a Normed Vector Space

Consider $\ell^\infty$ the vector space of real bounded sequences endowed with the sup norm, that is $||x|| = \sup_n |x_n|$ where $x = (x_n)_{n \in \Bbb N}$. Prove that \$B'(0,1) = \{x \in l^\infty ...
1answer
108 views

### Confused by proof in Rudin Functional Analysis, metrization of topological vector space with countable local base

I'm working through Rudin's Functional Analysis, and I am confused by a step in his proof for Theorem 1.24, which states that if X is a topological vector space with a countable local base, then there ...
1answer
106 views

### Connected components that are relatively open in $\sigma(T)$

Let $T$ be an bounded linear operator on a Banach space $X$. Suppose the spectrum of $T$, $\sigma(T)$ has infinitely many connected components, then $\sigma(T)$ must contain infinitely many ...
1answer
164 views

### Constructing a countable family of seminorms in a metrizable LCS.

Here's some context before my question. Let $\mathbb{V}$ be a topological vector space, which is Hausdorff and such that its topology is generated by some arbitrary family of seminorms ...
1answer
249 views

### Finding the topological complement of a finite dimensional subspace

I know that for any finite dimensional subspace $F$ of a banach space $X$, there is always a closed subspace $W$ such that $X=W\oplus F$, that is, any finite dimensional subspace of a banach space is ...
1answer
194 views

### Uniqueness of the derivative in locally convex topological vector space

I need a hint of proof of uniqueness of the derivative in locally convex topological vector space (it's asserted in Lang's "Introduction to differentiable manifolds"). Define derivative of a function ...
1answer
186 views

### Why This Map is Closed?

Consider the following definition of closed maps, defined in the book Nonlinear Programming by Bazaraa et al.: Let $X$ and $Y$ be nonempty closed sets in $\mathbb{R}^p$ and $\mathbb{R}^q$, ...
1answer
146 views

### Finest topology on a space of banach space operators

let $X$ be some Banach space. Let $L(X)$ be the set of continuous operators on $X$ into $X$. Let $(\tau_i)_i$ be a set of topologies on $L(X)$ s.t. $L(X)$ is topological vector space (i.e. addition ...
1answer
211 views

### Is any Banach space a dual space?

Let $X$ be a Banach space. Is there always a normed vector space $Y$ such that $X$ and $Y^*$ are isometric or isomorphic as topological vector spaces (that is, there exists a linear homeomorphism ...
1answer
56 views

### Openness of linear mapping 2

I quote a previously asked question : Let $X$ be a topological vector space over the field $K$, where $K=\mathbb{R}$ or $K=\mathbb{C}$, and let $\mathbb\{f\colon X\rightarrow K^n\}$ (\$n \in ...
1answer
121 views

### Existence of balanced neighborhoods in a topological vector space

I'm wondering about the following: Let$\ X$ be a topological vector space. Then one could pick balanced neighborhoods$\ W$ and$\ U$ of$\ 0$ such that $\ \overline{U} + \overline{U} \subset W$, ...
1answer
229 views

### How to define the derivative of Radon measures

Let $M$ be the positive borel measures on a hausdorff topological space $X$, which are finite on compacts sets $--$ i.e. the real cone of radon measures. I am given a definition of a derivative of ...
0answers
104 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 ...
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