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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62 views

Looking for a collection of applications of the Open mapping theorem

I am looking for various applications of the open mapping theorem (https://en.wikipedia.org/wiki/Open_mapping_theorem_(functional_analysis) ) other than only the closed graph theorem and that if two ...
3
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1answer
90 views

Regarding “stronger” norms

Let $X$ be a normed linear space. Show that a norm $\|\cdot\|_{1}$ is stronger than a norm $\|\cdot\|_{2}$ if and only if for any sequence $\{x_{n}\} \subset X$, $\|x_{n}\|_{1} \to 0$ always implies $\...
4
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1answer
62 views

How to prove that for a nonempty convex subset $S \subset X$ ($X$ is normed vector space) it is true that $\partial \overline{S} = \partial S$?

I am having trouble with the concept of convexity. This is the statement I'm trying to prove. Let $X$ be normed vector space. If $S \subset X$ is convex and $S^\circ \neq \emptyset$ then $\partial \...
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1answer
35 views

Show that $E$ is closed.

Let $X$ be a normed linear space .Let $T_n$ be a sequence of continuous linear operators on $X$ such that $\sup_n \|T_n\|<\infty$. Let $E=\{x:T_n x $ is Cauchy$\}$. Show that $E$ is closed. My ...
4
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1answer
48 views

showing $\|x\| \leq \lim_{n\to \infty} \inf \|x_n\|$

showing $$\|x\| \leq \lim_{n\to \infty} \inf \|x_n\|$$ where $x_n \to x$ weakly, and we are working under a normed space. I am given a hint that $$\|x\| = \sup_{\|\phi\| = 1} |\phi(x)|$$ where $\phi \...
2
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1answer
36 views

If two norms are not equivalent, then we have a sequence $\|x_n\|/\|x_n\|' \to 0$.

Definition 2.22. We say that two norms $\|\cdot\|$ and $\|\cdot\|'$ are equivalent if there exist $C_1,C_2>0$ such that $$C_1\|x_1\|' \le \|x\| \le C_2 \|x\|',$$ for all $x\in V$. If two norms $\|...
0
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1answer
72 views

In a normed set the boundary of a subset is contained in the boundary of the closure of the set.

Let $X$ be a normed space and $N$ a convex subset of $X$ (also $N^\circ \neq \emptyset$). I am trying to show that $\partial \bar N = \partial N$. I found the proof that $\partial \bar N \subset \...
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1answer
47 views

Is $C[a,b]$ complete when given the norm $||f||:=\sup_{x\in [a,b]} \Big|\int_a^xf(t)dt \Big|$?

Is $C[a,b]$ complete when given the norm $||f||:=\sup_{x\in [a,b]} \Big|\int_a^xf(t)dt \Big|$ ?
2
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2answers
43 views

If every absolutely convergent series is convergent then $X$ is Banach

Show that A Normed Linear Space $X$ is a Banach Space iff every absolutely convergent series is convergent. My try: Let $X$ is a Banach Space .Let $\sum x_n$ be an absolutely convergent series ...
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1answer
39 views

Normed space related question

Let $p$ be in the range of $0<p<1$, and consider the space $ L_p[0,1]$ of all functions with $$ \|x\| = \left[\int_{i=0}^1 |x(t)^p| \, dt\right]^{1/p} <\infty$$
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2answers
47 views

Can $X \ne \{0\} \implies X^*\ne\{0\}$ be proved without Hahn-Banach theorem?

We know that if $X \ne \{0\}$ is a NLS then $X^*\ne\{0\}$ ; is there any way to prove it without using Hahn-Banach theorem ? Thanks in advance
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1answer
34 views

Application of Linear Combination Theorem

In $(\mathbb{R}^{2},\|\cdot\|_{p})$ with $1 \leqslant p < \infty$ and under the standard basis $\{e_{1},e_{2}\}$ find the largest possible $c_{p} > 0 $ satisfiying $$ \|a_{1}e_{1} + a_{2}e_{2} ...
1
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0answers
18 views

$Y$ be a linear subspace of a NLS $X$ and $z \in X$ , then is it true that $dist (z,Y)=\sup \{f(z):f \in X^*,||f||=1,f(Y)=0\}$ ? [duplicate]

Let $Y$ be a linear subspace of a NLS $X$ and $z \in X$ , then is it true that $dist (z,Y)=\sup \{f(z):f \in X^*,||f||=1,f(Y)=0\}$ ?
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0answers
15 views

Does there exist an infinite dimensional real NLS , all whose proper linear subspaces are closed? [duplicate]

Does there exist an infinite dimensional real NLS $X$ all whose proper linear subspaces are closed ? I can only conclude one thing that if such an $X$ exist then it cannot be complete . Please help. ...
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0answers
32 views

Reverse triangle inequality in a normed linear space [duplicate]

I need to prove the following equation $$\lvert\lVert x\rVert-\lVert y\rVert\rvert \le \lVert x-y\rVert$$ How can I prove that? I used triangle inequality. But, get stuck.
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0answers
24 views

Show that $l^2(\Bbb N , F)$ with equipped norm is not complete.

For $v$ in $l^2(\Bbb N , F)$ with norm on $l^2$ defined as: $\lvert\lvert v\rvert\rvert_{W}= \sum^\infty_{k=1}\frac{\lvert v_{[k]}\rvert}{2^k}$ Show that $l^2(\Bbb N , F)$ with the norm $\lvert\...
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1answer
31 views

Determine whether a sequence space is normed

Let $(a_k)$ be a monotonously decreasing sequence of positive numbers with $a_1 =1$ and $a_k\to 0$. Let also $\sum_{k=1}^\infty a_k$ diverge. If $1\leq p<\infty$, show that $$A := \lbrace (x_k) : ...
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2answers
49 views

Is the dual space of a separable normed space also separable?

If $X$ is a real normed space such that $X^*$ (the dual) is separable then $X$ is also separable. Is the converse true, i.e., if $X$ is separable then is its dual space $X^*$ necessarily separable?
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2answers
52 views

Does every finite dimensional subspace of any normed linear space have a closed linear complement?

Let $Y$ be a finite dimensional subspace of a real NLS $X$ and let $Y=span \{y_1,...,y_n\} $ ; then by Hahn-Banach theorem , we can find continuous linear functionals $l_1,...,l_n \in X^*$ such that $...
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2answers
81 views

Proof of a p-norm and Cauchy/Cauchy-Schwarz inequality generalization

I've started to study metric spaces recently and got intrigued with a demonstration which led me to a bunch of ideas and questions. First of all, I've proved the Cauchy inequality and then Cauchy-...
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1answer
40 views

Convergent series which is not absolutely convergent in $(X,\|\cdot\|)$

Let $(X,\|\cdot\|)$ be a normed linear space. Recall from prior results that $(X,\|\cdot\|)$ is Banach $\iff$ any absolutely convergent series in $(X,\|\cdot\|)$ converges. (a) Give an example of ...
2
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1answer
37 views

Functional Analysis(Normed Spaces)

Let $X$ be the space of all complex valued square Riemann- integrable functions on $[0,1]$ with $2$- norm. Define the map $F:X\to X$ by $F(u)=v$ with $v(t)=\int\limits_{0}^{t}{ u^2(s) ds}$, then ...
10
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2answers
152 views

Regarding linear independence on a normed-linear space given a condition

Let $(X,\|\cdot\|)$ be a normed linear space and $x_{1}, x_{2}, \ldots, x_{n}$ be $n$ linearly independent vectors in $X$. Show that there exists $\epsilon > 0$ such that if $y_{1}, y_{2}, \cdots, ...
4
votes
1answer
51 views

A question in the proof of $C(X)$ is not a dual space of a Banach space.

Let $X$ be a non-singleton compact connected space.I want to show that $C_{\mathbb{R}}(X)$ is not the dual space of a Banach space,$\mathbb{R}$ is real number field. I already know that, extreme ...
2
votes
1answer
56 views

Prove that there exists NO norm such that $ f_n $ converges to $f$ iff $f_n$ converges to $f$ on compacta.

More specifically: Given the space of continuous functions $\mathbb{R}\rightarrow\mathbb{R}$ where convergence is equivalent to uniform convergence on compacta (i.e., compact sets), prove that ...
2
votes
1answer
225 views

Regulated function-higher dimensions

Let $\mathbb{X}$ be a Banach space and let the function $f:[0,1]^2 \rightarrow \mathbb{X}$ be continuous. (a) Show that $\forall\ \epsilon>0\ \exists\ \delta>0$ s.t. $\|f(x)-f(y)\|<\...
4
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1answer
50 views

How to show that $BV[0,1]$ ( the space of all functions on $[0,1]$ of bounded variation ) is not complete under supremum norm?

How to show that $BV[0,1]$ ( the set of all functions of bounded variation ) is not complete under supremum norm , by explicitly constructing a Cauchy sequence which does not converge or by ...
3
votes
2answers
52 views

Normed vector space with a closed subspace

Suppose that $X$ is a normed vector space and that $M$ is a closed subspace of $X$ with $M\neq X$. Show that there is an $x\in X$ with $x\neq 0$ and $$\inf_{y\in M}\lVert x - y\rVert \geq \frac{1}{2}\...
1
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1answer
23 views

Inequality problem in normed space involving maximum

Let $X$ be a normed space. Show that for every $x,y\in X$: $$\|x+y\|\leq \max\{\|x\|,\|x+2y\|\} $$ Wanted to check two cases. First, assume the maximum is $\|x+2y\|$, then $$\|x+y\|\leq \|x\|+\|y\|\...
2
votes
1answer
66 views

Minimizing a funtional in the Sobolev space $H_0^1$

I am trying to show that, given $f \in H^{-1}(U)$, there exists a unique $u \in H_0^1(U)$ such that: $$\int_U \nabla u\cdot\nabla v \, \mathrm{d}x= \langle f,v \rangle_{H^{-1}} \, , \quad \forall \, v ...
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3answers
29 views

Elements of a space

If A is a normed linear space and there exists an element x,y in A, then is it also true that x-y is also in A as well? I am pretty sure this is true by definition, but I am not able to find any ...
0
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2answers
41 views

When we raise $f$ to a positive power, what happens to the norm?

I am missing something in the identity $(1.18)$ below. What does that identity have to do with the fact that the $L^p$ norm is a non-negative number? The identity $(1.18)$ says that when we raise $f$...
0
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0answers
20 views

Applying homogeneity for a norm to prove triangle inequality

I am hoping to have someone help explain to me the steps of this proof of the triangle inequality for $L^p$ spaces, $p\ge 1$. Specifically the first two applications of the homogeneity property don't ...
4
votes
0answers
39 views

When is a topology is defined for this special kind of uniform convergence?

Question: Let $\phi:X\to X$ be bijective and continuous. Does a topology $\tau$ exist such that $$f_n\overset\tau\to f\Leftrightarrow \phi\circ f_n\overset\infty\to\phi\circ f$$ where $\infty$ ...
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1answer
68 views

If $u\in L^2(U)$ and $u>0$, how to show $u\ln u\in L^2(U)$?

$U$ is a bounded open subset of $R^n$. If $u\in L^2(U)$ and $u>0$, how to show $u\ln u\in L^2(U)$ ?
2
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0answers
16 views

Introduction to a textbook on Minkowki spaces

I want to learn more about the metric spaces, specially the "Minkowski spaces" and "Zermelo navigation problem" on Minkowski spaces. I have just studying the book " Riemann-Finsler Geometry" by ...
1
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2answers
94 views

showing a space with sup norm is complete

show that $(c_0,||\cdot||_\infty)$ be the space of real valued sequences converging to $0$ with the supremum norm is complete Okay so I know that the normed space is complete if any Cauchy sequence ...
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0answers
41 views

Triangle inequality in norms

Let $N:V\to\mathbb{R}$ be a sequence of norms and $d(x,y)=\sum_{j=1}^{\infty}2^{-j}\frac{N_{j}(y-x)}{1+N_j(x)+N_j(y)}$ Show that $d(x,z)\leq{d(x,y)}+d(y,z)$ I tried the following $d(x,z)=...
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1answer
38 views

$\sim$ is an equivalence relation on the set of norms on $X$.

Let $X$ be a vetor space. Prove that $\sim$ is an equivalence relation on the set of norms on $X$. Where $\sim $ is the equivalence of two norms. This seems very abstract. What exactly do I have to ...
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1answer
62 views

bounded functions, norms

Instead of answering this question could someone possibly explain what I need to do? I don't fully understand what the question is asking, firstly $M$ has been fully defined so how do we know $|M|$? ...
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1answer
66 views

Prove $\|\cdot\|_p$ and $\|\cdot \|_q$ aren't equivalent on $\ell^p$

$1 \leq p < q < \infty$ So we need to find something in $\ell ^p$ that gives different results in each of the two norms but I can't think of anything. I could think of something that is n $\...
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1answer
38 views

Prove $\| \cdot \|_{\infty}$ is well-defined on $\ell ^1$

$1 \leq p < q \leq \infty$ $p=1$ and $q= \infty$ Let $(a_k)_{k \geq 1} \in \ell^1$, then $$\sum_{k \geq 1}|a_k|^1< \infty$$ (*) which means $\sup_{k \geq 1 } |a_k|^1 < \infty$. Hence it is ...
2
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1answer
38 views

$X,Y$ be normed-linear spaces , $Y$ finite dimensional , $T: X \to Y$ be a non-continuous linear map , then is $\ker T$ dense in $X$?

Let $X,Y$ be normed-linear spaces ( over $\mathbb R$) , $Y$ is finite dimensional and $T: X \to Y$ be a linear map which is not continuous ; I know that then $\ker T$ is not closed in $X$ ; my ...
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1answer
50 views

Prove $\sum \frac{\sin(n^2 t)}{n^2}$ converges

Let $(X,\|\cdot\|)$ be a Banach space and a sequence $(x_n) \subseteq X$ be such that the series $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb R$. Prove that the sequence $S_n=x_1+\cdots+x_n$ ...
2
votes
2answers
50 views

Functional calculus for unitization of an algebra?

I have been stuck on this problem for a week now: "Let $A$ be a Banach algebra without identity, let $a\in A$, and let $f$ be holomorphic on a neighborhood of $\sigma(a)$, so that $f(a)\in A^\# $ is ...
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1answer
32 views

Prove $\sup _{t \in [0,1]} |P(t)|$ is a norm

Let $X$ be the vector space containing all polynomials with real coefficients. For every $P \in X$, define $N_1 (P)=\sup _{t \in [0,1]} |P(t)|$ I am having trouble proving the part when you show $||P|...
0
votes
1answer
45 views

An open ball is an open set by the continuity of the norm

I Hvev an exercise that I should prove that in a normed space, an open ball is an open set, but using the property of the continuity of the norm. Given a normed space $X$ and the norm $||.||$ such as ...
1
vote
0answers
15 views

Norms on matrices over Banach algebras

Let $A$ be a unital Banach algebra. Is there a sequence of norms $||\cdot||_n$ such that $(M_n(A),||\cdot||_n)$ is a Banach algebra for each $n$, identity matrices of all sizes have norm 1, the ...
4
votes
1answer
69 views

$X,Y$ infinite dimensional NLS , not both Banach , then $\exists T \in \mathcal L(X,Y)$ such that $R(T)$ is not closed in $Y$?

Let $X,Y$ be infinite dimensional normed-linear spaces , not both Banach , then does there necessarily exist a continuous linear transformation $T:X \to Y $ such that $range (T)$ is not closed in $Y$ ?...
2
votes
1answer
53 views

Alternative formulation of the supremum norm?

The supremum norm is defined as $$\|f\|_\infty=\sup\limits_X|f|$$ This induces a topology: $$ \begin{align}f_n\overset\infty\to g&\Leftrightarrow \sup\limits_X|f_n-g|\overset{\mathbb{R}}\to 0\\ &...