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.

learn more… | top users | synonyms

10
votes
1answer
966 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 ...
5
votes
3answers
520 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.
4
votes
1answer
288 views

An explicit construction for a “doubly weak” topology

Let $(X,s)$ be a topological vector space over $\mathbb{F}$ with linear topology $s$, which we will henceforth refer to as the strong topology. Then, as usual we can construct the continuous dual ...
4
votes
2answers
294 views

Continuous linear functionals

Let L be a continuous linear functional on a metric linear space X. Prove: L(S) is a bounded set for any bounded subset S of X. The metric is translation invariant.
4
votes
1answer
345 views

What is the topological dual of a dual space with the weak* topology?

I'm trying to understand a claim I heard in class. To be concrete, suppose $X$ is a compact, hausdorff topological space, and let $C(X)$ be the space of continuous functions on $X$ with the supremum ...
0
votes
1answer
234 views

When is $\| f \|_\infty$ a norm of the vector space of all continuous functions on subset S?

Let S be any subset of $\mathbb{R^n}$. Let $C_b(S)$ denote the vector space of all bounded continuous functions on S. For $f \in C(S)$, define $\| f \|_\infty = \sup_{x \in S} |f(x)|$ When is this a ...
0
votes
1answer
130 views

About the filtering family of seminorms

We first start with following definitions. Definition 1. A family $\mathcal{P}$ of seminorms on a real vector space $X$ is called filtering if for any $p_1,p_2\in \mathcal{P}$ there exsist $q\in ...
10
votes
1answer
356 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 $X$ ...
6
votes
0answers
142 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 $E_1$, ...
2
votes
1answer
162 views

Questions regarding internal and interior points for a convex subset of a topological vector space

Suppose that $X$ is a topological vector space, with a convex subset $A$. How do we show that if the vector $u$ is in the interior of $A$, then $u$ is an internal point of $A$ and if the interior of ...
-1
votes
1answer
184 views

Locally convex space which is not Banach.

I know that all Banach spaces are (Hausdorff)locally convex spaces. I would like to verify that the converse is not true by giving an example of a space which is locally convex but not Banach. I am ...
1
vote
2answers
57 views

Prove $\overline{x+A} = x+\bar{A}$ and $\alpha\bar{A} = \overline{\alpha A}$

If $A$ is a subset of $(V,\parallel.\parallel)$, then let $\bar{A}$ denote its closure. Show that if $x\in V$ and $\alpha \in \mathbb{R}$, then $\overline{x+A} = x+\bar{A}$ and $\alpha\bar{A} = ...
0
votes
1answer
102 views

show $\langle f,g \rangle _w = \int^b_a f(x)g(x)dx$ is an inner product

Let $w(x)$ be a strictly positive continuous function on [a,b]. Define a form on $C[a,b]$ by the formula $\langle f,g \rangle _w = \int^b_a f(x)g(x)dx$ for $f,g \in C[a,b]$. Show that it is an inner ...
1
vote
0answers
48 views

Show that the norm of the derivitive of a $C^1$ function over a vector space is non-negative, homogeneous and satisfies the triangle ineq

For $f$ in $C^1[a,b]$, define $p(f)= \parallel f'\parallel _{\infty}$. Show that $p$ is non-negative, homogeneous, and satisfies the triangle inequality. Why is it not a norm? -I can easily show the ...
3
votes
2answers
209 views

Left topological zero-divisors in Banach algebras.

Let $ A $ be a unital Banach algebra. Define $ \zeta: A \longrightarrow [0,\infty) $ by $$ \forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|, $$ where $ ...
1
vote
3answers
343 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 ...
2
votes
2answers
131 views
0
votes
1answer
199 views

Problem 5. ( chap3. p.87, functional analysis, W.Rudin)

I had done part a, b, and d,. But i cannot breakthrough part c, and part e,. I restate entired problem in the following: For $0<p<\infty$, let $l^p$ be the space of all functions $x$ (real or ...
2
votes
1answer
59 views

About the continuity of $B$ (problem 12 chap.2, p.55, functional analysis, W.Rudin)

Let $X$ be the normed space of all real polynomails in one variable, with $||f||=\int_0^1 |f(t)|dt$. Put $B(f,g)=\int_0^1 f(t)g(t)dt$, and show that $B$ is a bilinear functional on $X\times X$ which ...
0
votes
1answer
469 views

Density and closedness of $C[0,1]$ in $L^\infty[0,1]$ in norm and weak-* topologies

With results: "For convex subsets of a locally convex space, a, originally( strongly) closed equals weakly closed, and b, originally (strongly dense equals weakly dense." Could you help me solve this ...
1
vote
3answers
846 views

Return an array of evenly distributed points on a sphere give Radius and Origin. [duplicate]

Given a sphere of radius $r$, and origin $x,y,z$ what is the simplest way I can generate an evenly distributed array of points on the sphere $(x_1,y_1,z_1),(x_2,y_2,z_2),\cdots(x_n,y_n,z_n)$. Note I ...
2
votes
1answer
313 views

A surjective linear map into a finite dimensional space is open

I'm in search of different proofs of the following proposition: $\bf{Proposition}$: Suppose $X$ and $Y$ be topological vector spaces, $\text{dim }Y<\infty$, and $\Lambda:X\to Y$ is a surjective ...
0
votes
1answer
124 views

Examples of $T_0, T_1, T_3, T_4$ and Hausdorff spaces

What could be simple examples of $T_0$, $T_1$, $T_3$, $T_4$ and Hausdorff ($T_2$) topological spaces?
6
votes
1answer
261 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 ...
1
vote
1answer
187 views

Barrelled space

A locally convex space is called Barrelled if each closed absorbing convex set is 0-neighborhood See. But i doubt that every absorbing set contains zero. Then is every LCV is barreled. I think, ...
1
vote
1answer
57 views

Single norm criterion

Let $E$ be a metrizable locally convex space whose topology is defined by an increasing sequence $\{p_n\}$ of seminorms. Show that the topology of $E$ can be defined by a single norm iff there ...
0
votes
0answers
75 views

Is a closed set in a TVS over $\mathbb{R}$ convex?

From Theory of Convex Structures by M. L. J. Van De Vel, on a set $X$, a topology and a convexity structure are said to be compatible, if the convexity structure is generated by the closed sets. The ...
1
vote
0answers
49 views

Two different opinions on whether a topological vector space is a uniform space

Van de Vel's Theory on Convexity Structures says a TVS is uniform iff it is locally convex: 3.10.1. Proposition. Let $X$ be a topological vector space, equipped with the standard convexity and ...
5
votes
0answers
172 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 ...
1
vote
0answers
53 views

Determining Similarity of Unit Vectors

I'm seeking for an injective piecewise continuous function $f:\mathbb S^n\rightarrow[0,1]$ where $\mathbb S^N$ is the set of vectors with $L_2$ norm equals $1$. The piecewise continuity requirement ...
6
votes
1answer
100 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? ...
2
votes
2answers
105 views

Closure of opening of closure in $\mathbb R^2$

My question is somehow related to Closure of the interior of another closure However, I go a bit further. I have a closed set $X\subseteq \mathbb R^2$ and $Y:=\operatorname{cl}\operatorname{int} X$. ...
5
votes
1answer
158 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 ...
5
votes
1answer
91 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 ...
1
vote
1answer
120 views

If every linear functional is continuous in $V$, is every linear functional continuous for $S\subseteq V$?

Suppose $V$ is a finite dimensional (real) topological vector space. The first lemma in these notes says that Every vector subspace of a tvs with the induced topology is a topological vector space ...
3
votes
2answers
209 views

Is there a dumbed down version of the open mapping theorem for finite dimensional real vector spaces?

I would like to understand why any surjective linear transformation between finite dimensional topological real vector spaces, each with the natural topology, is in fact an open map. Reading around, ...
0
votes
0answers
53 views

Linear Application that is open in a TVS

Let $T: E \to F$ be a linear map between topological vector spaces $E$, $F$. If for each nonempty open set $G$, the interior of $T(G)$ is non-empty, then, $T$ is open. Proof: $$\mathrm{Int}(T(G))= ...
0
votes
1answer
43 views

Linear Application in a TVS

Let $E$ a Topological Vector Spaces, $T: E \longrightarrow \mathbb{K}$, linear. If for some $x \in E$, $Tx \neq 0$. Then, $T$ is open. I think that it is sufficient to prove that $T(G) \subset ...
2
votes
0answers
37 views

Density problem

$U$ is any open set of $\mathbb{R}$. We known that $C_0^\infty(U)$ is dense in $C^k(U)$. But what about, say $C_0^\infty((0,1))$ in $C^k([0,1])$?
2
votes
2answers
350 views

Closure of interior and interior of closure in a topological vector space

If $Y$ is a subset of topological vector space $X$ and is compact and convex show that $\overline{Y^\circ} = \overline{Y}$ and $\overline{Y}^\circ = Y^\circ$. I tried this way but I am not sure: ...
2
votes
3answers
185 views

Are vector subspaces of $\mathbb{R}^n$ always closed?

Suppose $S$ is any proper vector subspace of $\mathbb{R}^n$. Is $S$ a closed set in the usual topology on $\mathbb{R}^n$? Geometrically, I think it is clear that $S$ must be closed in $\mathbb{R}^n$ ...
4
votes
1answer
107 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 ...
1
vote
1answer
37 views

Continuity and openess in quotient space

The setting: $X$ and $Y$ are topological vector spaces. $N \subset X$ is a closed subspace. $T(N)=\{0\}$ $\pi : X \rightarrow X/N$ the quotient map. $S : X/N \rightarrow Y$ uniquely determined by ...
1
vote
1answer
74 views

Set boundary preserved by an infinite union

Suppose I have a subset $U\subset\mathbb R^2$ and a real number $r>1$ with the following properties: $U$ is compact; $U\subset rU$ (self-similarity); $0\in U$; there exists an open set $H\subset ...
0
votes
1answer
140 views

Triangle inequality of a metric on a quotient space of a topological vector space

In "Functional Analysis" by Rudin, a metric $\rho$ on the quotient space $X/N$ of a topological vector space $X$ and a closed subspace $N$ is defined as follows: For $x,y \in X$, $$ \rho ...
5
votes
1answer
553 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 ...
0
votes
2answers
79 views

Non-barreled topology compatible with the duality

Given $(X,s)$ a (real) barreled locally convex space (that is, every closed convex and absorbing set in $(X,s)$ is a neighborhood of the origin), is there a (strictly) finer, non-barreled linear ...
3
votes
0answers
105 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 ...
1
vote
1answer
235 views

Locally Convex Space Via Seminorms

Suppose that we have a Hausdorff locally convex space with its topology $\tau$ and let $P(X)$ be a separating family of $\tau$-continuous semi-norms so that $\tau$ is generated by $P(X)$. How do we ...
5
votes
0answers
131 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 ...