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.

learn more… | top users | synonyms

3
votes
2answers
44 views

Linear functional: continuous at $x_0=0 \iff$ continuous at all $x$

Let $f$ be a linear functional on a normed space $(X, ||\cdot ||)$. Prove that $f$ is continuous $\iff$ it is continuous at all $x \in X$. Backward direction is trivial: Since $f$ is continuous at ...
1
vote
0answers
38 views

On Banach space , is every linear bounded projection map an open map?

Let $X$ be a Banach space and $P \in \mathcal B(X)$ be a projection ( i.e. $P^2=P$ ) . Is it true that $P$ is an open map in the sense that for every open set $U$ in $X$ , $P(U)$ is open in $P(X)$ ? ...
0
votes
1answer
15 views

$V$ be a real vector space ; does every seminorm on it comes from the absolute value of some linear functional on $V$?

Let $V$ be a vector space over $\mathbb R$ and $p:\mathbb V \to [0,\infty)$ be a seminorm (http://mathworld.wolfram.com/Seminorm.html ) on it ; then is it true that there exist a linear transformation ...
4
votes
1answer
29 views

On existence of invariant subspace of continuous linear operator on Banach space such that $\{S(x): S \in (T)'\}=X $ for some $x$

Let $X$ be a Banach space , $T$ be a continuous linear operator on $X$ such that $\exists x \in X$ such that $\{S(x): S \in (T)'\}=X $ , where $(T)'$ is the commutant of $T$ , then I can show that ...
1
vote
1answer
31 views

Proving Corollary to Riesz's Lemma

Let $(X,\|\cdot\|)$ be a normed linear space and $Y \leqslant X$ be a proper subspace. If $\text{dim}(Y) < \infty$, show that there exists $x \in X$, with $\|x\| = 1$ such that $d(x,Y) = 1$. ...
0
votes
1answer
52 views

An everywhere discontinuous function

As usual, $\mathbb R[x]$ denotes the vector space of polynomials in one variable with real coefficients. It is easy enough (and a good exercise for beginners) to prove that the function ...
1
vote
1answer
45 views

$Y$ be real NLS ; if there is a Banach space $X$ such that there is a continuous linear open mapping from $X$ to $Y$ then is $Y$ a Banach space?

Let $Y$ be a real normed linear space ; if there exist a Banach space $X$ such that there exist a continuous linear open mapping from $X$ to $Y$ then is $Y$ a Banach space ?
0
votes
1answer
26 views

$X$ be Banach space , $T \in \mathcal B(X)$ be an open map , $Y$ be a closed linear subspace of $X$ ; is the restriction of $T$ on $Y$ an open map?

Let $X$ be a Banach space , let $T$ be a continuous open linear map from $X$ to $X$ , let $Y$ be a closed linear subspace of $X$ , then is $T_o$ , the restriction of $T$ on $Y$ , is an open map ? ...
2
votes
0answers
60 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
votes
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 ...
4
votes
1answer
61 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 ...
1
vote
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
votes
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
votes
1answer
32 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 ...
0
votes
1answer
71 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 ...
0
votes
1answer
40 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
votes
2answers
36 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 ...
1
vote
1answer
38 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 $$ \|x\| = \left[\int_{i=0}^1 |x(t)^p| \, dt\right]^{1/p} <\infty$$
1
vote
2answers
43 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
1
vote
1answer
24 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} + ...
1
vote
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 $dist (z,Y)=\sup \{f(z):f \in X^*,||f||=1,f(Y)=0\}$ ?
1
vote
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. ...
0
votes
0answers
30 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.
0
votes
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 ...
1
vote
1answer
29 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) : ...
1
vote
2answers
34 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?
0
votes
2answers
43 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 ...
1
vote
2answers
54 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 ...
1
vote
1answer
36 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
votes
1answer
35 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 ...
4
votes
1answer
43 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
55 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
222 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. ...
4
votes
1answer
38 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
50 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 ...
1
vote
1answer
21 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 ...
2
votes
1answer
63 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 ...
1
vote
3answers
28 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
votes
1answer
59 views

integral equation, bounded, norm

to be honest i'm not too sure where to start? i'm assuming I have to estimate the norm in the hint using the supremum norm in order to estimate $k(s,t)$ and then somehow use this information to ...
0
votes
2answers
36 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 ...
0
votes
0answers
19 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 ...
3
votes
0answers
37 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 $\infty$ ...
1
vote
1answer
66 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
votes
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
vote
2answers
92 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 ...
1
vote
0answers
38 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 ...
0
votes
1answer
34 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 ...
1
vote
1answer
59 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 $|M|$? ...
1
vote
1answer
65 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 ...
1
vote
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 ...