A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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5
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2answers
308 views

If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $

I have this problem I'm working on. Hints are much appreciated (I don't want complete proof): In a normed vector space, if $ x_n \longrightarrow x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} ...
3
votes
1answer
87 views

What are the $n$-th degree minimal polynomials for $L^p([-1,1])$?

It is known (even by me) that the Chebyshev polynomial of degree $n$ (of the first kind) is the minimal polynomial in the space $L^{\infty}([-1,1])$ for a fixed $n$ and leading coefficient $2^n$. ...
1
vote
0answers
64 views

Dense property of $C^k_0(\Omega)$

When $\Omega = \mathbb{R}^d$ or $\mathbb{R}^d_+$, $C^k_0(\bar{\Omega})$ is dense in $W^{k,p}(\Omega)$(Sobolev space, k-differentiable and each term estimated by p-norm). I am wondering if it holds ...
6
votes
1answer
857 views

Inequalities in $l_p$ norm

I'm having difficulty with the following problem. Any help would be appreciated. Problem: Consider the sequence spaces $l_p$ with the usual norm. If $1\le p\le q\le \infty$, I want to show the ...
1
vote
2answers
392 views

Inverse of continuous linear operator

This is a follow-up to this question. Part 3 of that question was stated incorrectly. The correct version is Show that a continuous invertible linear operator on a normed space has a continuous ...
4
votes
1answer
285 views

Normed Linear Space: Why does $\|x_n\|\to 0$ imply $x_n \to 0$?

I can prove the contrapositive: $x_n$ does not tend to $0$ implies either: $x_n$ diverges (does not converge), in which case neither does $\|x_n\|$, or $x_n$ converges to $x \neq 0$ which implies ...
5
votes
1answer
183 views

Compact operator norm estimate

I found the next exercise in Haim Brezis's book Functional Analysis, Sobolev Spaces and Partial Differential Equations. I feel like I solved the problem, but I'm not sure. The problem is: Let ...
2
votes
3answers
974 views

Interior of a Subspace

There is a conjecture: "The only subspace of a normed vector space $V$ that has a non-empty interior, is $V$ itself." (here, the topology is the obvious set of all open sets generated by the metric ...
1
vote
2answers
343 views

Norm-preserving map is linear

How can one show that a norm-preserving map $T: X \rightarrow X'$ where $X,X'$ are vector spaces and $T(0) = 0$ is linear? Thanks in advance.
1
vote
2answers
166 views

Closure of a nontrivial normed vector subspace that is equal to the whole space

Can you show me an example of a normed vector subspace $S$ strictly included in a normed vector space $V$ whose closure is equal to the whole $V$?
0
votes
1answer
186 views

How to find/parameterize vector perpendicular to circle of constant $\ell_p$ norm

This should be very easy, but I can't get my head around it: given $1\leq p < \infty$, and a point x with $\|x\|_p = 1$, how do I get a (or the unit, or any) vector which is perpendicular to the ...
0
votes
0answers
93 views

distance of an affine subspace to a polytope

I wonder how to prove the following statement. Let $V$ be a $d$-dimensional normed space with $d \geq 3$, let $P \subset V$ be a $(d-2)$-dimensional polytope. Then there is an $\epsilon > 0$ such ...
1
vote
1answer
358 views

Two weird proofs about continuity in normed vector spaces

I am reading a pair of "proofs" that a friend sent to me. I really don't understand some passages, so I hope someone could help me. The questions are the following First Question. The result to be ...
3
votes
3answers
165 views

Maximal Value of Integral

Calculate the maximal value of $\int_{-1}^1g(x)x^3 \, \mathrm{d}x$, where $g$ is subject to the conditions $\int_{-1}^1g(x)\, \mathrm{d}x = 0;\;\;\;\;\;\;\;$ $\int_{-1}^1g(x)x^2\, \mathrm{d}x = ...
12
votes
5answers
283 views

Passing from induction to $\infty$

Somehow, the operation of passing to the limit after I have shown that something is true by induction for each natural number $n$ troubles me each time. I know there are instances where one cannot ...
3
votes
3answers
85 views

$\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$ for functions in $C([0,1])$?

Why does the following hold for continuous functions on $[0,1]$? $\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$
0
votes
0answers
49 views

existance of the interpolation space

Let $X\subset L_1+L_2$ and let $Y$ be interpolation space between $L_2$ and $L_{\infty}.$ Given $U:X\longrightarrow Y$. My question is the following: Is there exists space $Z\subset Y$, such that ...
4
votes
1answer
146 views

$T(V)$ is a closed subspace of $V$?

Let $V$ be a normed vector space (not necessarily a Banach space) and let $S$ and $T$ be continuous linear transformations from $V$ to $V$. If we assume that $T=T \circ S \circ T$. Then how to show ...
2
votes
1answer
113 views

Limit inferior taken on the norm of a sequence

Let $E$ a normed vector space and let $(x_n)$ be a sequence in $E$. Suppose that $x_n$ converges weakly (i.e. wrt the weak topology) to $x$. Why is it that from the inequality $$ |f(x_n)| \leq \|f\| ...
0
votes
1answer
55 views

Finding a vector in a n.l.s.

Let $X$ be a normed linear space and $Y$ a closed proper subspace. Prove that for all $\varepsilon > 0$, there is an $x \in X$ with $\|x\| = 1$ and such that $\|x − y\| ≥ 1 − \varepsilon$ for all ...
2
votes
1answer
191 views

Norm closure of convex hull of its set of extreme points

How to prove that the set of extreme points of $B_{\ell^1} = \{v \in \ell^1 : \| v \| \le 1\}$ is $\{ +e^N, -e^N : N=1,2,3,\ldots \}$, where $e^N$ denotes the Nth standard basis element in $\ell_1$: ...
8
votes
2answers
1k views

$T$ is continuous if and only if $\ker T$ is closed

Let $X,Y$ be normed linear spaces. Let $T: X\to Y$ be linear. If $X$ is finite dimensional, show that $T$ is continuous. If $Y$ is finite dimensional, show that $T$ is continuous if and only if $\ker ...
2
votes
2answers
1k views

Prove that $X'$ is a Banach space

I'm taking a new course on functional analysis and meet with the following problem. If $X$ is a normed space (not necessarily complete), then prove that $X'$ is a Banach space. Definition: When the ...
2
votes
0answers
84 views

Consequence of metrizability proof - disregard, the question is an error

In Marian Fabian et al's Functional Analysis and Infinite-Dimensional Geometry, Proposition 3.22 states/proves that if $X$ is a separable Banach space, then the (closed) unit ball, $B_{X^{*}}$ of ...
1
vote
4answers
341 views

Hahn-Banach theorem (second geometric form)

I found in Brezis' Analyse Fonctionelle the Hahn-Banach theorem ("second geometric form") and there is a passage I can't understand. In newest versions of this book the proof has been modified. ...
5
votes
1answer
372 views

Norm for continuous linear functionals, newbie questions

Let $E$ be a normed vector space and let $f\colon E \to \mathbb{R}$ be a continuous linear functional. Define the dual norm of $f$ as $$ \|f\| = \sup_{\|x\|\leq 1} |f(x)|. $$ First question. I ...
1
vote
0answers
54 views

Disjointness of stars in a simplicial complex in $\ell_2$

Definitions Let's consider a full $n$-dimensional simplicial complex $C$ in $\ell_2(X)$. By that I mean a set of all functions $f:X \to[0,1]$ such that $\sum_{x\in X}f(x)=1$ and there are at most ...
2
votes
1answer
570 views

Every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$?

We have just come across the lemma that every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$. My question is that do all norms on $\mathbb{R}^n$ take the form $\|\cdot \|_1, \|\cdot \|_2, ...
2
votes
0answers
120 views

norms on a vector space - is there a quicker way to approach this problem?

I'm looking at the following problem, which I have done but wonder whether there isn't a faster way of doing it. (It's a past exam question which is supposed to take 7.5 minutes, but I only managed to ...
4
votes
1answer
177 views

There exists an isometric embedding

Let $W$ be a closed linear subspace of a normed vector space $V$. Let $i_V: V \to V^{**}$. and $i_W: W \to W^{**}$ be the canonical embeddings of V and W into their second duals. Prove that there ...
-1
votes
1answer
224 views

strictly convex space ---> strictly convex function

How would you prove that in a strictly convex normed vector space, the function $f(x) = \| x \|^2$ is strictly convex?? FYI: $E$ is strictly convex iff $\| t x + (1-t) y \| <1$ for all $x,y \in ...
2
votes
2answers
569 views

Coercivity vs boundedness of operator

The definition of coercivity and boundedness of a linear operator $L$ between two $B$ spaces looks similar: $\lVert Lx\lVert\geq M_1\lVert x\rVert$ and $\lVert Lx\rVert\leq M_2\lVert x\rVert$ for some ...
1
vote
0answers
321 views

Convergence of $L^p$ norms

Given a measure space $X$ with its measure $\mu$, it can be shown (I'll provide a proof if asked for) that $\displaystyle \forall f \in L^\infty(X,\mu),~\textrm{if } \exists p_0:\forall q \geq p_0, ...
1
vote
1answer
628 views

A normed space is locally compact iff its closed unit ball is compact.

To prove that A normed space is locally compact if and only if its finite dimensional, I need to prove a lemma: A normed space is locally compact if and only if its closed unit ball is compact. One ...
3
votes
2answers
256 views

Homogeneous function in bounded mean oscillation BMO($\mathbb R^n$) space

Let me recall some notations: The mean oscillation of a locally integrable function $u$ (i.e. a function belonging to $ L^1_{\textrm{loc}}(\mathbb{R}^n))$ over a cube Q in $\mathbb R^n$ (which has ...
2
votes
2answers
87 views

Mean value of convergent series

Let us in a normed linear space have a sequence $\{a_i\}_{i=1}^\infty$ which converges to some value $b$, how can I show that $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\frac{a_i}{n}=b$$ My idea is to use ...
2
votes
1answer
270 views

Hahn-Banach. Extend the functional by continuity

Let $E$ be a dense linear subspace of a normed vector space $X$, and let $Y$ be a Banach space. Suppose $T_{0}\in\mathcal{L}(E,Y)$ is a bounded linear operator from $E$ to $Y$. Show that $T_{0}$ can ...
1
vote
1answer
178 views

Definition of completeness and convergence in norm

Let $X$ be a normed space. We say that $X$ is complete if every Cauchy sequence in $X$ converges to an element of $X$ in norm. Now, in proofs of completeness, we start with $\{x_n \}$ Cauchy, and it ...
2
votes
2answers
104 views

Showing a function is not continuous in the one-norm

I have the following question: Consider the normed space $(C([0,1]),||\cdot||_{\infty})$. Let $x_{0}\in [0,1]$ and define $F:C([0,1])\to \mathbb{R}$ by $F(f)=f(x_{0})$ Show that $F$ is ...
2
votes
1answer
194 views

Continuity with normed spaces

Consider the normed space $(C([0,1]),||\cdot||_{\infty})$. Define $F:C([0,1])\to C([0,1])$ by $F(f)=f^2$ Show that $F$ is continuous with respect to $||\cdot||_{\infty}$. I've attempted ...
3
votes
1answer
572 views

How to show convergence in a metric space?

Suppose that $\{x_n\}→x$ where $\{x_n\}$ is a sequence in a normed space V and $x ∈ V$. Show that $\forall y ∈ V, \{x_n + y\} → x + y$.
2
votes
1answer
76 views

Question about normed spaces

Let $(X,||\cdot||)$ be a complete normed space. Let $F_1, F_2, F_3,\ldots\subseteq X$ be closed, non-empty subsets of $X$. Assume that $F_1 \supseteq F_2\supseteq F_3\supseteq \cdots$ and ...
2
votes
2answers
106 views

Complete normed spaces

Let $(X, ||\cdot||)$ be a complete normed space. Let $||\cdot||$ be a norm on $X$, and assume that there are constants $c_{1}$, $c_{2} \in (0,\infty)$ such that: $c_{1}||x-y||\le||x-y||_{0}\le ...
0
votes
2answers
118 views

Problem in normed spaces

Some help with the following would be great. Let $(X,||\cdot||)$ be a normed space. Let $(x_{n})_{n}$ and $(y_{n})_{n}$ be Cauchy sequences in $(X, D)$. Say also that $s_{n} = ||x_{n} + ...
4
votes
2answers
195 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 ...
37
votes
1answer
941 views

Banach Spaces - How can $B,B',B'', B''', B'''',B''''',\ldots$ behave?

(ZFC) Let $ \big\langle B,+,\cdot, \:\: \|\cdot\| \:\: \big\rangle $ be a Banach space. Define $ \mathbf{B} \; = \;\big\langle B,+,\cdot, \:\: \|\cdot\| \:\: \big\rangle $. Define $\: \mathbf{B}_0 ...
5
votes
1answer
2k 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 ...
1
vote
1answer
179 views

properties on normed vector space

Let $X \neq \{0\}$ a normed vector space.Prove the following (a) $X$ does not have isolated points. (b) If $x,y \in X$ such that $ ||x-y||= \epsilon >0$ then 1.Exists a sequence $(y_n)_n$ in $X$ ...
4
votes
2answers
330 views

How convergence relates to equivalence of norms

Let $X$ be a normed linear space with two norms $||\cdot||_1$ and $||\cdot||_2$. Prove or disprove that this statements are equivalent: $||\cdot||_1$ and $||\cdot||_2$ are equivalent, $\{x_n\}$ ...
4
votes
1answer
163 views

Notation: $L_p$ vs $\ell_p$

$L_p$ is often used to describe a norm, or a vector space with that norm (see e.g. wikipedia). Is $\ell_p$ (typically, or canonically) a different notation for the same concept, or is it used to ...