1
vote
1answer
42 views

What kind of space is this: $\Bbb{R}^n\times\Bbb{S}_{++}^n$?

Let $\Bbb{R}^n$ be the Euclidean space of $n$-dimensional column vectors with real coefficients. Moreover, $\Bbb{S}_{++}^n$ be the space of symmetric positive definite $n\times n$ real matrices. We ...
3
votes
0answers
52 views

Condition on Equality of closure of open ball and closed ball, suppose I have a counterexample

Let $(X, d)$ be a metric space. Also for $x \in X$ and $r \ge 0$ define: $$ B(x,r) = \{ y \in X : d(x,y) < r \} \quad \mbox{ and } \quad K(x,r) = \{ y \in X : d(x,y) \le r \}. $$ Denote by ...
1
vote
1answer
34 views

Proof of compactness for sets of norm equal to one in finite-dimensional normed vector spaces

The proposition I have been trying to prove is that the set $A=\{x\in E:N(x)=1\}$ is a compact subset of the (real) finite-dimensional vector space $E$ for any norm $N:E\to \mathbb{R}$. I am reading ...
6
votes
1answer
122 views

Radon-Riesz & Kadec-Klee

Let us say that a normed vector space has the a) RR (Radon-Riesz) property if for any sequence, norm convergence is equivalent to weak convergence + convergence of norms. b) KK (Kadec-Klee) property ...
0
votes
2answers
52 views

continuity in the strong topology implies continuity in the weak one

I have to prove that if $T:(E,\|\cdot\|_E)\rightarrow (F,\|\cdot\|_F)$ is a continuous and linear operator, and $x_h\rightharpoonup x$ in $E$, than $Tx_h\rightharpoonup Tx$ in $F$. So we know that $T$ ...
2
votes
0answers
26 views

Extension of function with values in a Banach space

I want to prove the following Let $E,X$ be Banach spaces, and $Y\subset E$ a closed subspace with codimension $1$. Let $T:Y \to X$ be a continuous linear function. Then there exists a continuous ...
0
votes
0answers
49 views

Show that an hyperplane is closed iff f is linear and continuous

I need an help with the following exercise. Let $(E,\| \cdot \|)$ a n.v.s. and let $f:E\rightarrow \Bbb R$. Show that $H=\{x\in E: f(x)=\alpha\}$ is closed if and only if $f\in E'.$ Actually, I ...
4
votes
3answers
65 views

Relation between continuous maps and convergence of sequences

I am studying metric spaces and I know that in a normed space $E$ a map $T:E \to E$ is contínuous if and only if $T(x_n) \to T(x)$ for every convergent sequence $x_n \to x$ in $E$. In my notes there ...
0
votes
0answers
44 views

Finding a homeomorphism between these two balls

Let $u_1,u_2,u_3 \in \Bbb C$ be the cubic roots of unity. Define two norms on $\mathbb{C}^2$, $$\Vert (x,y) \Vert_1 = \sqrt{\vert x \vert^2 +\vert y \vert^2} \ \text{and} \ \Vert (x,y) \Vert_2 = ...
2
votes
1answer
64 views

Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
2
votes
3answers
61 views

Let $V $be a vector space. Prove/Disprove: There is a norm $\|\cdot\|$, such that all subsets of $V$ are open sets in $(V,\|\cdot\|)$.

The Assignment: Let $V$ be a vector space over $\mathbb{R}$ with $V \not= \{0\}$. Prove or disprove: There is a norm $\|\cdot\|_d$ on $V$, such that all subsets of $V$ are open sets in ...
1
vote
1answer
57 views

Weak convergence on Banach space

Let $(X,|.|)$ be a Banach space, and $Y$ is a linear subspace of $X$ which is dense in $X$. Now if we have another norm $|.|_Y$ in $Y$ which is comparable to $|.|$ by $$|y|\leq C |y|_Y, \ \ \ \forall ...
1
vote
1answer
61 views

Topology: Open, Closed Set and infinity norm

Since last week I've been learning a bit about Topology in Calculus and know the basic definitions of open, closed, norm, etc. Now I try to solve this question but I don't know how to. Its really ...
1
vote
1answer
51 views

Subspaces and convergence in weak* topology

I would like to ask some questions regarding convergence in the weak* topology and subspaces. Let $X$ be a normed space with subspace $A \subset X$. Assume $X$ is endowed with the weak* topology. ...
0
votes
1answer
72 views

Separability of a normed space

The following is an Exercise 12, page 75 of conway's Functional Analysis. Let $\oplus_\infty X_i = \{x\in \sqcap X_i: ||x||=\sup||x(i)||<\infty\} $ where each $X_i$ is a normed space for $i\in I$. ...
0
votes
1answer
63 views

Covering of closed unit ball with closed balls.

Notations and definitions Let $E$ be a finite dimensional vector space with norm $||\;||$. Let $B$ denote the closed unit ball in $E$ and $B_r[a]$ the closed ball centered at $a$ with radius $r$. ...
1
vote
2answers
57 views

a functional analysis question

$X$ is a banach space and $f$ a non zero linear functional. I'm trying to show $null(f)$ not dense in X $\implies f$ continuous. I've tried a few approaches but I think the following seems the most ...
0
votes
0answers
25 views

Is the set, $\{f\in Y: \left\|f\right\|_{Y} \leq C \ \text{and} \ \left\|(|f|)\right\|_{X}\leq C \}$, closed in $(Y, ||\cdot||_{Y})$?

Put, $X= \text{The space of "nice" complex valued functions on } \mathbb R; \text{that is}, f:\mathbb R \to \mathbb C;$ so that $X$ is Banach space with respect to the norm ...
3
votes
1answer
189 views

Normed Vectors Spaces

Let $(E,\| \cdot \|_E)$ and $(F,\| \cdot \|_F)$ be two normed vector spaces over $\mathbb{C}$ and let $u: E\rightarrow F$ be a linear map. (a). Prove that the following conditions are equivalent: i. ...
2
votes
2answers
62 views

Topology induced by Scalar Product

I just had an idea: It is clear that every scalar product induces a norm and that a metric and that finally a topology. Turning this argument around we know: Not every topology induces a metric only ...
0
votes
1answer
204 views

Unit balls in normed spaces.

Assume we talk about the $n$ dimensional vector space over the reals. It is easy to see that for any norm the unit ball is a convex symmetric set. And here is my question : Let $A$ be a bounded , ...
1
vote
3answers
101 views

Is there a norm on ${\Bbb R}^{\Bbb N}$

Let $E={\Bbb R}^{\Bbb N}$ be the real vector space of real sequences. 1) Is there a norm on $E$? 2) Is there a norm $N$ on $E$ such that the restriction of $N$ to $\ell^2$ is finer than the ...
1
vote
1answer
127 views

Closure of $B=\{ f\in C'[0,1] : |f(x)|\leq 1, |f'(t)|\leq 1 \forall t\in[0,1]\}$ (NBHM $2005$)

If $A$ is the closure in $\mathcal{C}[0,1]$ of the set $B$ where $$B=\{ f\in C'[0,1] : |f(x)|\leq 1, |f'(t)|\leq 1 \forall t\in[0,1]\}$$ Then Which of the following is true? $A$ is closed. $A$ ...
1
vote
1answer
50 views

About the closedness of Banach space

I'm reading a proof in trying to prove that a Banach space $X$ is reflexive if and only if $X^{*}$ is reflexive. There's a point in the proof saying that $X$ is a closed subspace of $X^{**}$, but I ...
3
votes
2answers
102 views

Completeness/Compactness of a subset in a normed linear space

Let $(X,\|\cdot\|)$ be the normed linear space consisting of the sequences $a=(a_n)_{n=1}^{\infty}$, for which the corresponding series $\sum_{n=1}^{\infty} a_n$ converges absolutely, with norm ...
1
vote
2answers
39 views

Negative exponential distance

Let $X := \left\{(a_k)_{k \in \mathbb N}, a_k \in \mathbb C\right\}$. Let $d\left( (a_k)_{k \in \mathbb N}, (b_k)_{k \in \mathbb N} \right) := e^{-u}$ with $u$ the smallest integer $k$ such that $a_k ...
2
votes
1answer
104 views

About open mapping and closed range theorem

I'm self-learning Functional Analysis in Rudin's book and found some following statement hard to understand. Hope someone can help me clarify this. 1) $X, Y$ are Banach spaces, $T \in B(X,Y)$, let ...
0
votes
1answer
72 views

Properties of subspaces of Normed Vector Spaces

How does it follow that a subset of a normed vector space cannot be open if it does not contain an open ball $B_{\epsilon}(0)$ where $\epsilon > 0$? I just want to confirm also that for normed ...
3
votes
1answer
97 views

Subspaces of a Topological Vector Spaces

I have a few questions about topological spaces which I am currently studying. First some definitions that I am using: Definition of subspace topology: Given a topological space $(X,\tau)$ and a ...
1
vote
2answers
66 views

Prove that $d_{1}$ and $d_{2}$ induce the same topology

Suppose $d_{1}(x,y) = |x-y|$, $d_{2}(x, y) = |\phi(x) - \phi(y)|$, where $\phi(x) = \frac{x}{1 + |x|}$. Prove that $d_{1}$ and $d_{2}$ are metrics on $\mathbb{R}$ which induce the same topology. ...
2
votes
1answer
68 views

Infinite line is closed in $\mathbb{R}^n$

I have been reading the book "Elements of the functional analisys", by Kolmogorov and Fomin. At the chapter of Normed Linear Spaces, page 73 to be precise, the author makes the following definitions: ...
1
vote
1answer
282 views

Is every normed vector space a metric space?

I was trying to prove that every normed space is a metric space, and the first three proprierties came natural. However, when faced with proving the triangle inequality I had a bit of problems. I ...
1
vote
1answer
60 views

A question about the quotient topology in normed linear spaces.

Say $M$ is a closed linear subspace of normed linear space $N$. The coset of the form $x+M, x\in N$ in the quotient space $N/M$ is defined by $$\|x+M\|=\inf\{\|x+m\|:m\in M\}$$ Let us consider the ...
0
votes
0answers
41 views

$E_1+E_2$ is open if both open?

if $X$ be a norm linear space and $E_1,E_2\subseteq X$ then $E_1+E_2=\{x+y:x\in E_1,y\in E_2\}$ is open if both open? is open if one is open and another is closed? closed if both are closed? I just ...
2
votes
1answer
259 views

How to proof homeomorphism between open ball and normic space

How can I prove that an open ball $B$ in a normed vector space $X$ is homeomorphic to $X$?
3
votes
1answer
666 views

Weakly compact implies bounded in norm [duplicate]

The weak topology on a normed vector space $X$ is the weakest topology making every bounded linear functionals $x^*\in X^*$ continuous. If a subset $C$ of $X$ is compact for the weak topology, then ...
3
votes
1answer
63 views

$K$ is weakly-compact $\Longleftrightarrow$ $\Pi(K)$ is weak*-compact

Let $X$ be a Banach space and $K\subset X$. $\displaystyle \Pi:X \longrightarrow X$** canonical injection $\Pi(x)(f)=f(x)$ How can we prove that: $K$ is weakly-compact $\Longleftrightarrow$ ...
1
vote
1answer
96 views

The topology on bounded sets in $X$** of pointwise convergence on $B$ is metrizable

Let $X$ be a Banach space. If $B\subset X$* is a norm-separable How can we prove that: The topology on bounded sets in $X$** of pointwise convergence on $B$ is metrizable. $X$*$=B(X,\mathbb{R})$ : ...
2
votes
1answer
566 views

Weakly closed implies sequentially closed

Another problem involving the weak topology: Let $X$ be a normed space and $A \subset X$ weakly closed. Then $A$ is sequentially closed, that is: If $(x_n) \subset A$ and $x_n \xrightarrow{w}x$, then ...
4
votes
1answer
95 views

Weak and strong topology on infinite dimensional spaces

Is there a simple example to show that the weak and strong topology on an infinite-dimensional space do not need to coincide? I have several ideas using differences in the weak and strong convergence ...
3
votes
3answers
74 views

One-point compactification of a linear space

Is it possible to take the one-point compactification of a linear space and get again a linear space? Like if $X$ is a normed space add $*$ to it and define $|x-*| = 1$?
8
votes
2answers
151 views

Two terms that I want to understand: weakest topology and jointly continuous (in the following context).

I was reading an article online, please help me to understand the following lines (in bold letters). - Topological structure: If (V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric and ...
4
votes
2answers
142 views

Why doesn't the Weierstrass approximation theorem imply that every continuous function can be written as a power series?

I hope that my question in the title is well formulated. I am a little bit confused with the next exercise from a book: Argue that there exists functions $f \in C[0, \frac{1}{2}]$ that cannot be ...
1
vote
1answer
118 views

Is the metric induced by a norm ''unique''?

Let $(X,\rVert{\cdot}\lVert)$ be a normed vector space. Clearly there is a canonical way to induce a metric (and so a topology) on $X$ by defining $d(x,y)=\rVert x-y\rVert$. Are there other metrics ...
2
votes
2answers
150 views

If $x\mapsto \| x\|^2$ is uniformly continuous on $E$, the union of all open balls of radius $r$ contained in $E$ is bounded $\forall r > 0$

A subset $E$ contained in $\mathbb{R}^n$ is such that the function $x \mapsto \left\Vert x\right\Vert^2$ is uniformly continuous on $E$. For $r > 0$, let $E_r$ denote the union of all open balls ...
3
votes
1answer
54 views

How to show $A=\{T\in \mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{T is onto}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$

How can we show $A=\{T\in \mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{T is onto}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$? Here $\mathcal{L}(\mathbb R^m, \mathbb R^n)$ is the set of ...
1
vote
1answer
95 views

Norms on a vector space over $\mathbb{R}$

In an exercise in the book "Topology and groupoids" the following is asked: Let V be a finite dimensional right vector space over $\mathbb{R}$ ($dim_RV=n)$. Show that any $2$ norms on $V$ are ...
6
votes
1answer
139 views

why must a normed space homeomorphic to a complete metric space be complete?

Why must a normed space X homeomorphic to a complete metric space Y be complete? I've solved this in the case where Y is a normed space (considering open balls gives Lipschitz equivalence), but am ...
10
votes
1answer
880 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 ...
2
votes
1answer
65 views

Is it true that non-equivalent norms share no nontrivial open sets?

Suppose $X$ is a normed linear space with two non-equivalent norms $||\cdot ||_1$ and $||\cdot ||_2$. It is clear that they must generate different topologies, but I was wandering if it were true that ...