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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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 ...
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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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23 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
51 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 ...
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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
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1answer
51 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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25 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} ...
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1answer
82 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 ...
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1answer
54 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 ...
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0answers
20 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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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.
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1answer
40 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
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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
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1answer
23 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 ...
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1answer
16 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 $\varphi : l_p \rightarrow \mathbb{R}$, $\varphi ((x_n)_{n \in \mathbb{N}})=\sum_{n=1}^{\infty} a_n x_n$ ...
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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.
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1answer
15 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 $B$ ...
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1answer
67 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 ...
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0answers
39 views

A question about parallelogram identity [duplicate]

Let $(V,\|\cdot\|)$ be a normed vector space. Then prove that $\|\cdot\|$ is from a scalar product if and only if $(\|u+v\|)^2+(\|u-v\|)^2=2(\|u\|^2+\|v\|^2)$ for every $u$ and $v$ from $V$. I don't ...
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1answer
35 views

Prove that $(V,\lVert\cdot\rVert_\infty)$ is a Banach space

$X=C[0,1]$ (i) Prove that if the sequence $(f_n)_{n\ge1} \subseteq X$ converges to $f \in X$ in the supremum norm, then for each $t\in[0,1]$ one necessarily has $\lim_{n\to\infty} f_n(t) = f(t)$. ...
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1answer
35 views

Prove that N is a norm on $ℓ^3$

For each $z = (c_n)_{n≥1} ∈ℓ^3$, let $$N(z) =\bigg( \sum _{n=1}^{\infty} \frac{|c_n|^3}{|n|^3} \bigg)^{1/3}$$ Prove that N is a norm on $ℓ^3$. You may use without proof standard facts. Sequences ...
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2answers
43 views

Is $c_{00}$ closed in $(\ell^\infty,\|\cdot\|_∞)$

Consider the normed space $(X,\|\cdot\|)= (\ell^\infty,\|\cdot\|_\infty)$ and its linear subspace $V = c_{00}$ consisting of all sequences $(a_n)_{n≥1}$ of real numbers that eventually become zero: ...
2
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1answer
29 views

Explanation of an exercise of Hahn-Banach in finite dimensional space

The problem is the following: Let $X$ be a finite-dimensional normed space. Prove that, if $A$ and $B$ are non-empty disjoint convex sets, there exists some hyperplane $H$ that separates $A$ and ...
1
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1answer
49 views

Orthogonality Relations Exercise, Brezis' Book Functional Analysis

I studying Brezis' book and I have somes partial solutions of the exercise $1.17.$ Let be $E$ a normed space and $f\in E^*$ be a linear functional nonzero. Consider the set $M=[f=0]$ given by ...
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1answer
24 views

A convex subset of normed vector space is path-connected

Let $(N, \|\;\|)$ be a normed vector space and $(X,\tau)$ a convex subset of $(N,\|\;\|)$ with its induced topology. Show that $(X,\tau)$ is path-connected, and hence also connected. What I have done ...
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110 views

Find distance to a set (subspace) without computing closest point

General setup: we have a finite-dimensional normed linear space $(V, \| \cdot \|)$, a subspace $U \subset V$, and a fixed vector $v_0 \in V$. We want to find the distance between $v_0$ and $U$. (No ...
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1answer
40 views

$\{x_n\}$ a sequence in NLS $X$ s.t. $\sum f(x_n)$ converges for all $f \in X^*$ , is the function $f \in X^* \to \sum f(x_n)$ continuous ?

Let $X$ be a NLS , $X^*$ be the set of all bounded real valued functions on $X$ ( i.e. the topological dual of $X$) , let $\{x_n\}$ be a sequence in $X$ such that $\sum_{n=1}^{\infty} f(x_n)$ ...
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1answer
109 views

Goldstine theorem and weak topology on $E$ induced by weak* topology on $E''$

In the book where I'm studying there this exercise: "(Goldstine's theorem). Recall that $E \subset E''$, or, more precisely, $J_E(E)$ is a subspace of $E''$ where $J_E :E \longrightarrow E''$ is ...
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 ...
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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 ...
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0answers
21 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 ...
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3answers
63 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 ...
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2answers
33 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 ...
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0answers
31 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 ...
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1answer
30 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 ...
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1answer
24 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 ...
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1answer
17 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)) = ...
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1answer
70 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 ?
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1answer
53 views

Detailed working of $||f||$

Given $\ell^{\infty} \rightarrow \mathbb R$ defined by $$f(a_1,a_2, a_3, ...)=\frac 1{\sqrt {0!}}a_1 + \frac {-1}{\sqrt {1!}}a_2 + \frac 1{\sqrt {2!}}a_3 +... +\frac {(-1)^{n-1}}{\sqrt {(n-1)!}}a_n$$ ...
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1answer
36 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 ...
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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 ...
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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 ...
2
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0answers
48 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: ...
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0answers
19 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 $x \in H$ ...
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3answers
47 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 ...
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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
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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
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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
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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
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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 ...