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.

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Show that, if we allow infinite intersection in the usual topology in R reduces to the discrete topology.

If X is a set and T is the collection of all the subsets of X, then all required of a topological space are automatically satisfied. This topology is called the discrete topology. Let X be the real ...
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Topology :definition of neighbourhood of a point and basic questions

I am going through the basics of topology, in order to deal with topological vector space. I haven't taken any course of topology so I have some fundamental questions (I 've seen only some topological ...
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Is a sequentially compact (non-metrizable) uniform space totally bounded?

First some topological definitions in terms of nets and sequences: A topological space $(X, \tau)$ is compact iff every net has a convergent subnet sequentially compact iff every sequence has a ...
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Open sets in the unitary group $U(\mathcal{H})$ of a Hilbert space $\mathcal{H}$.

Let $H$ be an infinite dimensional Hilbert space and let $(x_i)_1^\infty$ be an orthonormal basis for $H$. Consider $U(H)$ the unitary group of the continuous unitary operators on $H$. Equip $U(H)$ ...
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embeddings of TVS are continuous

How to show that the two embeddings of TVS are continuous: $C^{\infty}_c\subset S\subset C^{\infty}$? According to Wikipedia, the definition of "Continuously embedded" is that " one normed vector ...
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Check proof that the embedding of the unit ball $B\subset X$ into $X^{**}$ in weak-* dense

I have to prove the following theorem: Let $X$ be a (real) Banach space, and let $B$ denote its closed unit ball, and let $\tau (B)$ denote its canonical embedding into $B^{**}$, the closed unit ...
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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 ...
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Existence of linear continuous functional on locally convex TVS

Let $c>1$ and $X$ be a locally convex TVS. Take $a,b\in X$ and a closed subspace $Y$ of $X$ such that $Y\cap \{(1-t)a+bt \ | \ t\in\mathbb{R}\}=\emptyset$. How to prove that there exist $f\in X'$ ...
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$V$ is finite dimensional iff $V'$ with the weak topology is normable

Why is the following statement valid? Note, $V$ is locally convex Hausdorff topological vector space over $\mathbb{C}$ and $V'$ is the space of all continuous linear maps from $V \to \mathbb{C}$. $V$...
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Locally bounded topological vector spaces

Let ‎$X$ ‎‎‎be a topological vector space ‎such that every neighborhood of zero contains an infinite dimensional subspace. Then ‎$X$ is not locally bounded. I do not know, why? We know ‎$X$ is ...
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Is Topological Space a Metric Space?

What's the correct relationship between these two spaces? I think that topological space is a metric space, since the open is defined by a metric such that $d(x, a) < \epsilon$.
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The weak topology on $H$ is the weak* topology on $H^*$ pulled back via $\Phi$

I'm reading the following in Analysis Now by Pedersen: The map, $H$ a Hilbert space $$\Phi:H\to H^*: x\mapsto(\cdot\mid x)=[y\mapsto (y,x)]$$ is a conjugate linear isometry. Then define the weak ...
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Does $\frac{1}{n}x_0$ converge to the origin in any topological vector-space?

Let $X$ be a topological $\mathbf{R}$-vector-space (not necessarily Hausdorff) and $x_0 \neq 0$ a non-zero element of $X$. Then intuitively the sequence $(\frac{1}{n} x_0)_{n \in \mathbf{N}}$ ...
I shall begin with some definitions. 1) Suppose that $X$ is a topological (additive) group and $(x_{s})\subseteq X$ is a net, we said that $(x_{s})$ is Cauchy whenever $U$ is a neighbourhood of $0$, ...
I am reading a book in which a theorem is only proven analogously in the situation of weak topology. The proof does not seem to work in the case of weak$^{*}$ topology. Let $X$ be a normed linear ...