Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [locally-convex-spaces]

For questions about topological vector spaces whose topology is locally convex, that is, there is a basis of neighborhoods of the origin which consists of convex open sets.

This tag has to be used with (topological-vector-spaces) and often with (functional-analysis).

537 questions
22
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7 answers

Is there such thing as an unnormed vector space?

I learned about Banach spaces a few weeks ago. A Banach space is a complete normed vector space. This of course made me wonder: are there unnormed vector spaces? If there are, can anyone please provide any examples? Some thoughts: A complete space…
Sujaan Kunalan
  • 11,194
21
votes
1 answer

Is $[0,1]^\omega$ homeomorphic to $D^\omega$?

Let $n\in \mathbb N$ and let $D^n$ be the closed $1$-ball in $\left(\mathbb R^n, \|\,\cdot\,\|_1\right)$. It is not too hard to show that $[0,1]^n \cong D^n$ in this case. This observation leads to the question whether we also have $[0,1]^\omega…
Sam
  • 16,398
16
votes
1 answer

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 $\mathcal{F}$ is actually a Banach space. I'd like to…
student
  • 4,027
13
votes
1 answer

Elementary applications of Krein-Milman

Recall that the Krein-Milman theorem asserts that a compact convex set in a LCTVS is the closed convex hull of its extreme points. This has lots of applications to areas of mathematics that use analysis: the existence of pure states in C*-algebra…
Paul Siegel
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13
votes
2 answers

Reference request: infinite-dimensional manifolds

The following books and/or notes develop various aspects of the theory of infinite-dimensional manifolds: Lang, Fundamentals of Differential Geometry. Kriegl & Michor, The Convenient Setting of Global Analysis. Choquet-Bruhat & DeWitt-Morette,…
mdg
  • 1,152
12
votes
1 answer

Supremum of Banach Spaces

Let $X$ be a linear space with a family of complete norms $(\| \circ \|_I)_{I \in \mathcal{I}}$ on $X$, i.e. for every $I \in \mathcal{I}$ the tuple $(X,\|\circ\|_I)$ is a Banach space. Now define $$\| x \|_\infty := \sup_{I \in \mathcal{I}} \| x…
x_Y_z
  • 283
12
votes
2 answers

Definition of the convolution with tempered distributions and Schwartz function

In the book where I'm studying there is the following exercise. If $x \in \mathbb{R}^n$, $\varphi \in \mathcal{S}(\mathbb{R}^n)$ and $u \in \mathcal{S}'(\mathbb{R}^n)$ we define $(u \ast \varphi)(x)=\langle \tau_x \widetilde{\varphi} , u \rangle$,…
user288972
  • 2,490
10
votes
1 answer

Krein-Milman theorem and dividing pizza with toppings

In this question the OP mentions the following problem as an exercise on Krein-Milman theorem: You have a great circular pizza with $n$ toppings. Show that you can divide the pizza equitably among $k$ persons, which means every person gets a piece…
Martin Sleziak
  • 56,060
10
votes
3 answers

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 $X \times X \rightarrow X, (x, y) \mapsto x + y$ and scalar-multiplication $\mathbf{R} \times X…
Steven
  • 4,851
9
votes
1 answer

How is the weak-star topology useful?

Today I learnt something about the weak-star topology, but I don't know what the use of weak-star topology is. I hope someone can tell me what we can do with the weak-star topology. Thanks in advance!
Brain Zhang
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9
votes
1 answer

Are weakly compact sets bounded?

Let $X$ be a Hausdorff locally convex topological vector space, and let $X'$ denote its topological dual, that is, the vector space of all continuous linear functionals on $X$. If $A$ is a weakly compact subset of $X$, that is, if $A$ is…
serenus
  • 945
9
votes
1 answer

Dual space of space of all smooth function

On the space $C^\infty(S^1,\mathbb R)$, for each $n\in \mathbb N$, define $$p_N(\gamma)= \max\{|f^{(k)}(t): t\in S^1, k\leq N\}$$ Topology of all norms above define a metrizable locally convex topology (in fact Frechet space) on this space [Rudin…
Junu
  • 805
9
votes
2 answers

Is the topology that has the same sequential convergence with a metrizable topology equivalent as that topology?

Let $\mathscr T_1$ and $\mathscr T_2$ be two topologies on space $X$. Assume that $(X,\mathscr T_1)$ is metrizable, and any sequence in $X$ that converges in one of the two topologies must also converge in the other topologies, i.e.,…
Dreamer
  • 2,102
9
votes
2 answers

Topology on the space of test functions

I try to read into the theory of distributions and there is one thing which bothers me. I read that a distribution is a linear, continuous functional from the space of test functions, which, depending on the author, is sometimes defined as the…
fweth
  • 3,800
8
votes
1 answer

Strong topology vs Natural topology

Let $X$ be a locally convex space and $\left< X, X^{\prime} \right>$ stands for the dual pair. The bidual of $X$ is denoted by $X^{\prime \prime}$ and this is a dual of $X^{\prime}$ with a strong topology $\beta(X^{\prime}, X)$, i.e. a topology on…
Edvin Goey
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