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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1answer
27 views

When is injective contraction isometry

Let $f:X \to Y$ be a injective linear map between (semi-)normed spaces, s.t $B_Y = f(B_X)$, $B_X,B_Y$ being the unit balls. Is $f$ an isometry? If so, was there a superfluous requirement? EDIT: I ...
0
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1answer
19 views

Equivalent particular norms, can you point the right direction?

Let $P([0,1])$ be the space of all the polinomial with complex entries, defined in $[0,1]$ . Show that $||f||_\infty=sup_{t\in [0,1]}|f(t)|$ and $||f||_1= \int_{0}^{1}|f(t)|dt$ are equivalent norms. ...
1
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0answers
30 views

Spectrum of Linear Operator

The spectrum of a linear operator on a finite dimensional space is pure point spectrum, that is, both continuous and residual spectrums are empty. Or we can say that on a finite dimensional space, ...
0
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2answers
27 views

Let $C$ be a convex closed nonempty subset of a Hilbert space $H$. Show that there is a unique element in $C$ with the minimum norm.

Let $H$ be a Hilbert space and $C \subset H $ be a convex, closed and nonempty subset of $H$. Prove that there exists a unique $x_0\in C$ with minimum norm among the elements of $C$. I don't know ...
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2answers
31 views

Product topology on a product space of normed spaces is normable iff the product is finite [duplicate]

Suppose $(X_{i}, \Vert \cdot \Vert_i)_{i\in I}$ are all normed spaces over the same field $\Phi= \mathbb{R}, \mathbb{C}$ and suppose $X= \prod_{i \in I} X_i$ is the product space. I want to show that ...
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0answers
28 views

Proving that these functionals are bounded, and finding their norms.

Proving that these functionals are bounded, and finding their norms. $$a.)f_1(x):c_o \to \mathbb R , f_1(x)=\sum_{n=1}^{\infty}\frac{x_n}{2^{n-1}} \\ b.)f_2(x):l_1 \to \mathbb R , ...
2
votes
1answer
42 views

$|v|_{2,\Omega}=0$ implies $v=0$

I am stuck on this computation: let $\Omega$ be a domain in $\mathbb R^2$ and let $\Gamma_0$ be a relatively open proper subset of $\Gamma:=\partial\Omega$. Define $$ V=\{v \in H^2(\Omega); ...
2
votes
1answer
32 views

Showing a certain map is a norm.

Define $$\|x\|=\sqrt[3]{(|x_1|^2+|x_2|^2)^{3/2}+|x_3|^3}$$ for any $x=(x_1,x_2,x_3)\in \Bbb R^3$. Show that $\|\cdot\|$ is a norm on $\Bbb R^3$. I stuck on showing triangle inequality. I don't know ...
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0answers
32 views

Finding $\|f\|$

Given $f: \ell^2 \rightarrow \mathbb R$ defined by $$f(a_1,a_2,...)=\sum_{n=1}^{\infty} \frac{(-1)^n}{3^{n-1}} a_n$$ Is $f$ a bounded linear functional? If yes, then find $\|f\|$. It is definitely ...
1
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1answer
52 views

Symmetric norms on $\mathbb{R}^n$ are between the $\ell^1$ and $\ell^\infty$ norms

We say that a norm $\|\cdot\|$ on $\mathbb{R}^n$ is symmetric if it is invariant under sign changes and permutations of the components. Suppose that $$\|(1, 0, ..., 0)\| = 1$$ for a symmetric norm. ...
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1answer
28 views

Clarification about completeness of metric spaces

This is probably a very silly question but it bothers me for some time. We define a metric space $X$ to be complete if every Cauchy sequence in $X$ converges to some point in $X$. But any metric ...
2
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0answers
22 views

Find value of $p$ such that $\sum |b_n|^p $ converges

Let $(X , \|\cdot \|)=(\ell^2, \|\cdot \|_2)$. Let $e_n \in \ell^2$. Fora bounded linear functional $\phi$ on $X$, let $b_n=\phi (e_n)$ for each $n \geq 1$. Find the value of $p \geq 1$ such that the ...
0
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0answers
19 views

Showing a subspace is closed

Let $X = (\textbf{c}_{0},\|\cdot\|_{\infty})$ and $Y$ be defined as $$ Y := \bigg\{ \{x_{i}\} \in \textbf{c}_{0} : \sum_{i=1}^{\infty} \frac{x_{i}}{2^{i}} = 0 \bigg\}. $$ (1) Show that ...
3
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2answers
380 views

Can the real vector space of all real sequences be normed so that it is complete ?

Let $X$ be the vector space of all real sequences . Does there exist a norm on $X$ which makes it complete ?
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1answer
34 views

Show that an open linear mapping between normed spaces is surjective

I'd just like to know where to begin. The exact thing to prove: Let $X$ and $Y$ be normed spaces and $R:X\to Y$ is an open linear mapping. Show that $R$ is surjective. And to be clear, neither of the ...
0
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1answer
17 views

$X,Y$ be real NLS ; $T:X \to Y$ be a linear map such that $\ker T$ is closed ; then does $T$ have closed graph?

Let $X,Y$ be real normed linear spaces and $T:X \to Y$ be a linear map with closed kernel ; then does $T$ have closed graph ? What if we assume arleast one of $X,Y$ to be complete ?
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2answers
37 views

On the dimension of a real Normed Linear Space possessing a certain property

Let $X$ be a real NLS such that for every proper subspace $Y$ of $X$ , $\exists x \in X$ such that $||x||=1$ and $dist (x,Y)=1$ ; then is $X$ finite dimensional ?
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1answer
12 views

$Y$ is a ( closed) proper subspace of a real NLS $X$ such that $dist (x,Y)=1$ for some $x \in X$ with $||x||=1$ ; is $Y$ finite dimensional?

Let $Y$ be a finite dimensional proper subspace of a real NLS $X$ , we know that we can find $x\in X$ ( depending on $Y$) , such that $||x||=1$ and $dist (x,Y):=\{||x-y||:y\in Y\}=1$ . I would like to ...
0
votes
2answers
21 views

$X$ be finite dimensional real NLS , let $x \in X$ , does there exist $T \in \mathcal B(X)$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$?

Let $X$ be a finite dimensional real normed linear space , let $x \in X$ , then does there exist a continuous linear transformation $T:X \to X$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$ ? ...
2
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0answers
20 views

$f \in \mathcal l^{\infty}{'} $ ; $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty} $ is a sequence of non-negative terms ; is $f$ bounded? [duplicate]

Let $f:\mathcal l^{\infty} \to \mathbb R$ be a linear functional such that $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty} $ is a sequence with non-negative terms ; then is $f$ continuous ?
5
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1answer
80 views

$f_1,…,f_n$ be linear functionals on a real vector space $V$, then is there a norm on $V$ which makes every $f_i$ continuous?

Let $V$ be a real vector space, $f_1,...,f_n$ be linear functionals on $V$; then does there exist a norm on $V$ with respect to which each of $f_i$ is continuous? And what if we have infinitely many, ...
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1answer
12 views

Let $X$ be a non-zero real NLS , $x,y \in X$ , $B(x,r)\subseteq B(y,s)$ , then is it true that $r \le s$ ?

Let $X$ be a non-zero real NLS , $x,y \in X$ , $B(x,r)\subseteq B(y,s)$ , then is it true that $r \le s$ ? If $y=x$ then it is easy to see that that's the case . So I thought let $y \ne x$ ; I tried ...
0
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0answers
11 views

Extended continuous linear transformation keeping same norm , when co domain is finite dimensional

Let $Y$ be a subspace of a real normed linear space $X$ , $T:Y \to \mathbb R^n$ be a continuous linear transformation ; then can we extend $T$ to a continuous linear transformation $\bar T : X \to ...
0
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1answer
15 views

$X$ be Banach space with a proper dense subspace $Y$. Can the identity operator on $Y$ be extended to a continuous function from $X$ into $Y$?

Let $X$ be any Banach space with a proper dense subspace $Y$. Can the identity operator on $Y$ be extended to a continuous function from $X$ into $Y$ ?
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1answer
21 views

Is it possible to extended finite rank continuous linear transformation to a continuous linear transformation with same range?

Let $X,Y$ be normed linear spaces , $W$ be a linear subspace of $X$ , let $T:W \to Y$ be a continuous linear tranformation with finite rank i.e. $T(W)$ is finite dimensional ; then can we extend $T$ ...
0
votes
1answer
20 views

Show $c_{00}$ is not closed under supremum norm

Show $Y=c_{00}$ is not closed under $(\ell^{\infty}, \|\cdot\|_{\infty})$. I know that I need to find a $(y_n) \in c_{00}$ such that this converges to $y$ with $y \notin c_{00}$. So we need $\|y_n ...
0
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0answers
16 views

Finding norms on a piecewise function

For each $n=1,2,...$ let the function $g_n \in C[0,1]$ be defined by \begin{equation} g_n(t)=\begin{cases} 2nt & 0 \leq t \leq 1/2n \\ 2-2nt & 1/2n \leq t \leq 1/n \\ 0 & 1/n \leq t \leq ...
2
votes
2answers
45 views

Linear functional is continuous $\implies$ it is bounded

Let $f:X \rightarrow \mathbb R$ be a continuous linear functional. Prove $f$ is bounded. Since it is continuous, $\forall \varepsilon >0$, there exists $\delta >)$ such that ...
5
votes
0answers
49 views

Proving linear operator is bounded

Prove that the formula $T(b_1,b_2,b_3,...,b_n,...) = (b_1, b_2/2 ,..., b_n/n ,...)$ defines a bounded linear operator $T : (ℓ^∞,∥·∥_∞)→(ℓ^∞,∥·∥_∞)$. Proving that it is linear is easy. Need help with ...
2
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3answers
166 views

Weak convergence $\iff$ strong convergence in finite dimensional space

I am seeking a proof of the following claim. Weak convergence $\implies$ strong convergence in a finite-dimensional normed linear space. Thank you.
0
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1answer
29 views

$X,Y$ be Banach , $T \in \mathcal B(X,Y)$ be onto ; then , for every sequence $y_n \to y \in Y$ , $\exists x_n \to x\in X$ s.t. $T(x_n)=y_n , T(x)=y$?

Let $X,Y$ be Banach spaces , $T:X \to Y$ be a surjective continuous linear transformation , then is it true that for every convergent sequence $\{y_n\}$ in $Y$ , converging to $y \in Y$ , there exist ...
0
votes
1answer
31 views

What are the differences between $l^p$ space and $L^p$ space?

I always thought small $l^p$ space, or so-called Banach space, is the space equipped with a norm similar to vector norm as $$||x||_p = \left( \sum_{i\in\mathbb{N}} |x_i|^p \right)^{\frac{1}{p}}$$ ...
0
votes
1answer
27 views

The image of linear operator, $T(\ell ^{\infty})$

$T:(\ell^\infty, \|\cdot\|_\infty) \rightarrow (\ell^\infty, \|\cdot\|_\infty)$ with $T(b_1,b_2,\ldots)=(b_1, b_2/2, b_3/3,\ldots)$ is a bounded linear operator. Show that $w = (1, 1/\sqrt2, 1 ...
0
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0answers
39 views

$Y,Z$ be linear subspaces of a Banach space $X$ ; $Y$ be finite dimensional , $Z$ closed in $X$ ; is $Y+Z$ closed in $X$? [duplicate]

Let $Y$ and $Z$ be linear subspaces of a Banach space $X$ , such that $ Y$ is finite-dimensional and $Z$ is closed in $X$ , then is $Y +Z$ also closed in $X$ ?
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0answers
24 views

The concept of Subspace of a Normed Vector Space

I am working with Banach Spaces, which are complete Normed Vector Spaces (NVS). The norm on a NVS $(E, ||\cdot||_1)$ defines a metric which in turn defines a topology. Now let us consider $F ...
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1answer
53 views

Isometry and isomorphism normed spaces

Problem. Let $X$, $Y$ be real normed vector spaces and $ f $ isometry space $ X $ in the space $ Y $. Show that there is isomorphism $ A $ spaces $ X $ on the space $ Y $ and vector $ c \in Y $ such ...
0
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0answers
20 views

$C[a,b]$ with $\|f\|:=\int_a^b |f(x)|d(x)$ is a normed space but not a Banach space [duplicate]

Let $-\infty\lt a\lt b \lt \infty$. Let $C[a,b]$ is space of continuous functions and a function $\|.\|:C[a,b]\to R$ is given by $\|f\|:=\int_a^b |f(x)|d(x)$ a). Show that $\|\cdot\|$ is a norm in ...
3
votes
1answer
55 views

Characterizing orthogonally-invariant norms on the space of matrices

Denote by $M_n$ the space of $n \times n$ real matrices. We say a norm on $M_n$ is orthogonal invariant if: $$\|OX \|=\| XO\|=\|X \| \, \, \forall O \in O_n,X \in M_n$$ I am trying to characterize ...
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vote
0answers
26 views

Find $\|B\|_{\ell^1 \rightarrow \ell^1} $

$B(a_1,a_2,...)=(a_1/1,a_2/2,\ldots, a_n/n,\ldots) $ This is a linear operator defined on $\ell^1$. Notice that $\|Bx\|=\sum |a_n/n| \le \sum |a_n| =\|x\|$. So $\|B\|=\sup_{\|x\|_1 \le 1} ...
1
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1answer
84 views

Normed Space Isometric to Proper Subspace of Itself

I am trying to show that there exists a normed vector space which is isometric to a proper subspace of itself. I have been playing around with the $l^\infty$ norm on $\mathbb{N}$, but am struggling to ...
-1
votes
1answer
63 views

The space $C[a,b]$ with $\|f\|:=\int_a^b |f(x)|d(x)$ is a normed space but not a Banach space

Let $-\infty\lt a\lt b \lt \infty$. Let $C[a,b]$ is space of continuous functions and a function $\|.\|:C[a,b]\to R$ is given by $\|f\|:=\int_a^b |f(x)|d(x)$ a). Show that $\|\cdot\|$ is a norm in ...
1
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0answers
23 views

Why is $x\in X$ a weak star continuous linear functional in the dual?

I am reading an excerpt from Infinite Dimensional Analysis by Aliprantis and on page 235 it claims that if $X$ is a normed space, then "$x$ is a weak* continuous linear functional by definition". ...
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votes
1answer
44 views

Is $(1,0,…,0,-n^2,0,0,…) \in \ell^2$?

I am a bit unsure of this because if $n$ is very large then the sum would not be finite but then again the term after the nth term is $0$ onwards.
0
votes
1answer
42 views

Show $\langle f,g\rangle$ is not an inner product

Let $X = C[−1,1]$ be the space of continuous functions $f : [−1,1] → \mathbb R$. For $f,g ∈ X$ define $$\langle f,g\rangle =\int_0^1 f(t)g(t)dt$$ If I choose $f(t)=-t$ and $g(t)=1$, then $\langle ...
3
votes
1answer
33 views

Is the result true when the valuation is trivial and $\dim(X)=n$?

Here I proved the following result: Proposition: Let $(K,|\cdot|)$ be a valued field with non-trivial valuation and let $X$ be a vector space over $(K,|\cdot|)$. Two norms $p_1,p_2$ on $X$ are ...
2
votes
1answer
26 views

Does a convergent sequence of norms of a vector space always converge to a norm?

I thought that just as the sequence of norms $||x||_p :\mathbb{R}^n \mapsto \mathbb{R}$ converges to $||x||_{\infty}$ maybe there is some result that proves that every convergent sequence of norms of ...
0
votes
1answer
17 views

Computing the norm of $\varphi ((x_n)_{n \in \mathbb{N}})=\sum_{n=1}^{\infty} a_n x_n$

Let $p,q>1$, $\frac{1}{p}+\frac{1}{q} = 1$ and $(a_n)_{n \in \mathbb{N}}\in l_q$. Show that $\varphi : l_p \rightarrow \mathbb{R}$, $\varphi ((x_n)_{n \in \mathbb{N}})=\sum_{n=1}^{\infty} a_n x_n$ ...
0
votes
1answer
34 views

“Adding” inner products

Is there a general rule to simplify things like $<x,y> - <x,z>$ or generally $<.,.> \pm <.,.>$ I cant find anything in my notes that talks about this.
1
vote
1answer
21 views

Normed spaces: Sum of closures is a subset of the closure of the sum

Let $E$ be normed space and $A,B\subset E$. Show that $ \overline{A}+\overline{B}\subset \overline{A+B}, $ where $A+B=\{ a+b :a\in A \text{ and } b\in B \}$. Now I know this is true if $A$ and $B$ ...
4
votes
1answer
71 views

Norm of linear combination of vectors in the “same general direction”

Let $X$ be a normed vector space (a Banach space if necessary) and $x, y \in X$ such that $||x|| \leq ||x + y||$ and $||y|| \leq ||x + y||$. (Intuitively, I take this to mean $x$ and $y$ are in the ...