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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2
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1answer
50 views

$X$ be Banach , $T:X \to \mathcal l ^{\infty}$ be linear , $(Tx)_n$ the $n$-th term of $T(x)$;$f_n(x)=(Tx)_n$ ; if each $f_n$ is bdd then so is $T$?

Let $X$ be a Banach space , $T:X \to \mathcal l ^{\infty}$ be a linear transformation , for each $x\in X$ and each $n \in \mathbb N$ , $(Tx)_n$ be the $n$-th term of $T(x)$ and for each $n \in ...
0
votes
0answers
31 views

Difference between $\ell ^{\infty}$ and $\ell^p$

With $p \in [1,\infty)$. With the $\ell^p$, the set is to do with summations but with the $\ell ^{\infty}$ it just says the supremum of a given vector right? Can someone explain why there is no sum in ...
1
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0answers
22 views

Cauchy Schwarz inequality on scalar terms

The following equations are derived from Quantum information theory, which requires the use of Cauchy Schwarz inequality for a proof. I am quite puzzled by the second term in the summation, which ...
1
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3answers
64 views

Prove $\int _a^b |f(t)| \, dt $ is a norm

Let $a < b$ be real numbers and $X = C[a,b]$ be the space of continuous functions $f : [a,b] → \mathbb R$. Prove that $$ \|f \|_1 =\int _a^b |f(t)|\,dt $$indeed defines a norm on $X$. Struggling on ...
0
votes
2answers
34 views

An open set of subspace is it open in the space?

$\mathcal M_n(\Bbb R)$ is endowed by some norm and let $\mathcal S_n(\Bbb R)$ the subspace of symmetric matrices. If we have $\mathcal O$ is an open set of $\mathcal S_n(\Bbb R)$ is it also an open ...
0
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0answers
32 views

Continuity of Busemann-Hausdorff area density

tl;dr Why is the Busemann-Hausdorff area density continuous? Note that I posted a slightly different question regarding this on MO as well. Let $V$ be an $n$-dimensional normed vector space and ...
0
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1answer
31 views

Please help how to show that $x_{n_k}$ is convergent

In a normed linear space $X$ if every absolutely convergent series is convergent then prove that the space is a Banach Space. My try: Let $x_n$ be a Cauchy Sequence in $X$ .Then we can find a ...
-1
votes
1answer
25 views

Linearity of a Given Functional [closed]

Let $f$ be a functional from the set of all sequences with only finitely many nonzero terms, $c_o$, to a field $\mathbb{F}$ (real or complex). $f$ is defined by a series from $n=0$ to infinity ...
1
vote
1answer
18 views

Norm of the Dual Transform = Norm of the Transform?

For a continuous (bounded) linear transformation $T$ between vector spaces $T: V \to W$ the dual transformation is defined between their continuous dual spaces as $T' : W' \to V'$ where $T'(w'(v)) = ...
2
votes
1answer
71 views

$f:\mathbb R \to \mathbb R^n$ be such that $G(f):=\{(x,f(x)):x \in \mathbb R\}$ is closed and connected in $\mathbb R^{n+1}$ , is $f$ continuous ?

Let $f:\mathbb R \to \mathbb R^n$ be a function whose graph $G(f):=\{(x,f(x)):x \in \mathbb R\}$ is closed and connected in $\mathbb R^{n+1}$ , then is $f$ continuous ?
1
vote
1answer
53 views

Detailed working of $||f||$

Given $\ell^{\infty} \rightarrow \mathbb R$ defined by $$f(a_1,a_2, a_3, ...)=\frac 1{\sqrt {0!}}a_1 + \frac {-1}{\sqrt {1!}}a_2 + \frac 1{\sqrt {2!}}a_3 +... +\frac {(-1)^{n-1}}{\sqrt {(n-1)!}}a_n$$ ...
1
vote
1answer
41 views

Showing that a bounded linear function that is identity on a certain subspace is identity e'erywhere

Let $X$ be a normed space, $M$ a closed subspace, and a quotient of $X$ over $M$ defined as an ordered pair $(Q, \pi)$ such that $Q$ is a normed linear space $\pi$ is a bounded linear function from ...
0
votes
1answer
57 views

Boundedness and norm of a linear operator

Consider the linear operator $T : C[-\pi,\pi] \to \mathbb{R}$ defined by $$ Tf := \int_{-\pi}^{\pi} f(t)\sin(t)\phantom{.}dt $$ Show that $T$ is bounded and find its norm $\|T\|$. Consider ...
0
votes
1answer
30 views

Check $F_2(f)=f'(1)+f(1)$ is a linear functional

Let $(X,\|\cdot \|)=(C[0,1],\|\cdot \|_{\infty})$ and $F_1 :X \rightarrow \mathbb R$ be defined by $$F_1 (f)=\int _{1/2}^{3/4} f(t) dt$$ $F_1$ is a continuous linear functional. Lets consider the ...
3
votes
0answers
50 views

What is this operator topology?

Let $X$ be a separable Banach space with (norm $1$) Schauder-Basis $\{e_n\}_{n\in\mathbb N}$. Denote for $x\in X$ with $|\cdot|_x$ the seminorm on $\mathcal L(X)$ given by $|A|_x = \|A x\|$. Consider: ...
0
votes
0answers
20 views

Show $L$ is a closed linear subspace of $H$

Let $H$ be a Hilbert space and linear operator $P:H \rightarrow H$ satisfy $<Px,y>=<x,Py>$ for all $x,y \in H$ and $P^2=P$. We can use the fact that $Px \perp (x-Px)$ for every $x \in H$ ...
1
vote
3answers
50 views

Show $Px \perp (x-Px)$ and prove $||P|| \leq 1$

Let $H$ be a Hilbert space and linear operator $P:H \rightarrow H$ satisfy $\langle Px,y\rangle = \langle x,Py\rangle$ for all $x,y \in H$ and $P^2=P$. (a) Show that $Px \perp (x-Px)$ for every $x ...
0
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0answers
33 views

Is this function a norm?

Let $p$ be a real number such that $p \geq 1$, let $a$ and $b$ be real numbers such that $a < b$, and let $X$ be the set of all the real- (or complex-) valued functions that are defined and ...
1
vote
1answer
32 views

Find $\|T\|_{\ell^\infty \rightarrow \ell^2}$ [duplicate]

Let $(v_n)_{n \geq 1} \in \ell ^2$ be a fixed bounded sequence of real numbers. Define a mapping $T$ on $\ell^\infty$ using the formula $$T(a_1, a_2,\ldots)=(v_1a_1,v_2a_2,\ldots), \qquad ...
1
vote
1answer
18 views

Show $||Tx||_2 \leq ||(v_n)||_2 \cdot ||x||_{\infty}$

Let $(v_n)_{n \geq 1} \in \ell ^2$ be a fixed bounded sequence of real numbers. Define a mapping $T$ on $\ell ^{\infty}$ using the formula $$T(a_1, a_2,...)=(v_1a_1,v_2a_2,...), \, \, \, \, \, ...
0
votes
1answer
21 views

Show $Tx \in \ell ^2$ for every $x \in \ell ^{\infty}$

Let $(v_n)_{n \geq 1} \in \ell ^2$ be a fixed bounded sequence of real numbers. Define a mapping $T$ on $\ell ^{\infty}$ using the formula $$T(a_1, a_2,...)=(v_1a_1,v_2a_2,...), \, \, \, \, \, ...
1
vote
2answers
31 views

Show $F_1$ is a continuous linear functional

Let $(X,||\cdot ||)=(C[0,1],||\cdot ||_{\infty})$ and $F_1 :X \rightarrow \mathbb R$ be defined by $$F_1 (f)=\int _{1/2}^{3/4} f(t) dt$$ Show $F_1$ is a continuous linear functional. So we need to ...
0
votes
3answers
31 views

Norm of a functional is a norm on a v.s. $X^{*}$

Prove that $\| \cdot \|_{X^{*}}$ is indeed a norm on $X^{*}$, the space of bounded linear functionals on a normed space $(X, \| \cdot \| )$. I am not sure what to do in this. I do know that we ...
3
votes
2answers
45 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
40 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
30 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
35 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
54 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
27 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
61 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
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 ...
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
33 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
43 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
37 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
46 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
25 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
37 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
45 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 ...