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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Exercise 1.64 of Megginson's “An Introduction to Banach Space Theory”.

Could someone verify whether my solution to the following exercise is correct? The reason I am a bit in doubt is because the chapter of which this Exercise is part consists of the Banach-Steinhaus ...
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43 views

How can higher dimension spaces have smaller unit balls? [duplicate]

I have recently been shown the gamma function and a few of its uses, and one of those is calculating the measure of the unit ball in $\Bbb{R}^n$. The formula shows the measure going to zero (rather ...
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33 views

Uniqueness of endpoints of half-open line segments in linear spaces.

I try to solve the following exercise, which is Exercise 1.18 in Robert Megginson's An Introduction to Banach Space Theory. Let $X$ be a linear space, and define for any $x_1, x_2 \in X$ the line ...
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28 views

Continuous operators

We have that $T:E\rightarrow \mathbb{R}$ is linear where $E$ is a normed space we have that $\ker T=\{x\in E, Tx=0\}=T^{-1}(\{0\})$ if we suppose that $\ker T$ is closed, as $\{0\}$ is closed can we ...
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25 views

Example to show that unconditional convergence does not imply absolute convergence in infinite-dimensional normed spaces.

I tried to show that unconditional convergence does not imply absolute convergence in infinite-dimensional normed spaces using a direct proof, but unfortunately I did not succeed. The definition of ...
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30 views

Space of bounded linear operators might fail to be a Banach space.

I tried to show that the space of bounded linear operators $B(X,Y)$, where $X$ and $Y$ are normed linear spaces, might fail to be a Banach space. To show this, I considered the space $X = \ell^1 (\...
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44 views

Show that there is a bounded linear functional $\ell : \mathscr C [0,1]\to\mathbb R$ with $\lVert \ell \rVert \leq 1,\ \ell(1)=0,\ \ell(\cos(x))=1$.

The title says it all. I've been assuming that the best way to do this is constructively, by finding such an $\ell$. I have by a theorem in our class that since $\lVert \cos(x)\rVert =1$, we know that ...
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36 views

Why is the following scaling good for the general case?

In the following post how come scaling to $x_0=0$ and $r=1$ can help to prove for any $r,x_0$? Is there a general rule for when is it OK to scale?
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58 views

Generalized Poincaré Inequality on H1 proof.

let's see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable ...
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26 views

Finding the closest element to a function in a normed space containing functions.

Let $B=\{f\in c[1,0]|\forall 0\leq x\leq 1 : f(x) \geq 0\}$. Given $f\in c[0,1]$, find the closest element in $B$ under the $\|\cdot\|_2$ norm. I can see something similar to this question in the ...
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150 views

Show that if $[Q,P]=it\Bbb{I}$ then the operators are unbounded

In the Hilbert space $\mathcal{H} = L^2(\mathbb{R},dx)$, let 2 symmetrical operators $P$ and $Q$ be given, with the following properties: $D(P) = D(Q) = \mathcal{S}(\mathbb{R})$ $P\mathcal{S}(\...
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64 views

Inequality on a general convex normed space

Assume $(X,\|\cdot\|)$ is a normed space with the following property: if $x \neq y \in X$ have norm 1 then $\|\frac{x+y}{2}\|<1$. (We then say that $X$ is strictly convex) Prove that if $C$ is a ...
3
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1answer
31 views

Norm equivalence on $l^1$.

Suppose that $\|\cdot\|$ is a norm on $l^1$ such that: a) $(l^1, \|\cdot\|)$ is a Banach space, b) for all $x \in l^1$ $\|x\|_{\infty} \leq \|x\|$. Prove that the norms $\|.\|$ and $\|.\|_1$ are ...
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31 views

$L^p$ Norm of product of two bounded functions

If $f$ and $g$ are bounded functions in $L^p[a,b]$, does the following inequality hold in $L_p$ spaces? $$\|fg\|_p\leq\|f\|_p\|g\|_p$$
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Why is $T\colon X \to Y$ a homeomorphism?

I am reading through a paper in which the following is stated without proof: If $X$ is a normed space with norm $\| \cdot\|_X$ such that every norm on $X$ is equivalent to $\| \cdot\|_X$ then the ...
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32 views

am I misunderstanding boundness of a transformation?

Let $X$ be a Banach's space $Y$ a normed vector space $H\in B(X,Y)$ a family of bounded linear transformations $X\rightarrow Y$ and $V_n:=\{x\in X:\exists T\in H$ such that $||Tx||>n \} n\in \...
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45 views

What does “extend to X linearly” mean?

I'd ask someone to edit my question, I'm not sure if it's correctly spelled. Let $X$ be a normed vector space with Hamel basis $\{e_n\}$,$n\in \mathbb{N}$ consistent in unit vectors $e_n$ and $Te_n=...
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1answer
20 views

Length of a curve under a non-Euclidean norm in the integral form.

Let $V$ be a normed space. Let $\gamma\colon [a,b] \rightarrow V$ be continuous. Then $\gamma$ is a curve. Let $P$ be a partition of $[a,b]$, then $$\Lambda(\gamma, P) := \sum_{i=1}^n \| \gamma(x_i) -...
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16 views

Which one is the correct definition of natural norm?

In the definition 2 of Normal Subgroup Reconstruction and Quantum Computation Using Group Representations, the authors have defined the natural norm of a matrix as follows. The natural norm of the ...
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32 views

Why the separate notation for norm

One usually denotes the norm as $\|\cdot\| $, $\| v\| := \sqrt{\langle v, v \rangle}.$ However, in metric spaces, one often writes $d(x,y) \equiv \lvert x-y \rvert$. Since the norm canonically ...
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35 views

Compactness in a vector space

If $E$ is a normed space and $F$ is a subspace of $E$, how to prove that if $F\neq\{0\}$ then $F$ is not compact? I begin by this let $x\in F$ then $F=\bigcup_{x\in F} B(x,\varepsilon)$ how to say ...
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For an open subset $D$ of a normed space, its multiple $\alpha D$ is also open

$D$ is a subset of $\mathbb R$ and $E=\{\alpha x : x\in D, \alpha>0\}$ Prove that $E$ is open iff $D$ is open For each $\alpha x \in E \exists$ a ball $B_\epsilon (\alpha x)\subset E$. Can ...
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48 views

Prove that this transformation inverse exists and it's bounded

If $X,Y$ are Normed Vectorial Spaces, $T$ is a bounded lineal transformation. Prove that if exists $b>0$ such that $\|Tx\|\geq b\|x\| \forall x\in X$. Then $T^{-1}:Y\rightarrow X$ exists and it's ...
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1answer
10 views

Proof about converging absolutely with respect to equivalent norm

If series converges absolutely with respect to some norm, then it also converges absolutely with respect to any kind of equivalent norm. I need to prove this assertion, but I have no idea from where ...
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9 views

Minimizing matrix norm via left-multiplication by $SL(m)$

Suppose that $M$ is an $m\times n$ matrix of full row rank, with $m \leq n$. Then if $\|M\|$ is the matrix norm induced on $M$ from the norm on our vector space, we can look for the following ...
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19 views

Is this question well-formed? Let $T\in\mathcal B(X,Y),\ \lVert T\rVert<1,\ Y$ Banach, show $\sum_{n=0}^\infty T^n\in\mathcal B(X,Y)$.

If $X=Y$, then I think I have solved this problem entirely already. But if $X\neq Y$, then I don't understand what is being asked. How is $T^n$ defined when $X\neq Y$? When I consider $X=Y$, then I ...
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28 views

prove the map is a contraction

Let $L$ be a fixed positive real number. Let $K:[0,T] \times \mathbb{R} \rightarrow \mathbb{R} $ be continuous and satisfy the lipschitz condition $|K(s,x)-K(s,y)| \leq L|x-y|$ for all $s \in [0,T]...
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29 views

Why is $\ker(T-\lambda I)^n$ finite-dimensional?

Let $X$ be a normed space and $T$ be a compact operator on $X$ and $\lambda \in \sigma(T)\setminus\{0\}$. A closed unit ball in $\ker(T-\lambda I)$ always admit a convergent subsequence of a sequence,...
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27 views

Closed convex hull = closure of convex hull?

If the "closed convex hull" of A is the intersection of all closed convex sets containing A, is this the same as the closure of the convex hull of A? Many have asked whether the closure of the convex ...
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Find $\langle f,g \rangle$ w.r.t. $L_0 \perp L_1$.

Let $X=C[-1,1]$, and $L_k= \{ <t^{k+2i}, i=0,1,2,... > \} $. Define an inner product on $X$ with respect to $L_0 \perp L_1$. Then confirm that $L_0 \perp L_1 $ on your inner product. Can we ...
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some detail calculation on the proof of equivalence of norms

We say that two norm $\|x\|_1$ and $\|x\|_2$ on a vector space $X$ are said to be equivalent if there exists $K>0$ and $M>0$ such that $$ K\|x\|_1\le \|x\|_2\le M\|x\|_1 $$ Prove that on a ...
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1answer
52 views

Find the norm of a linear continuous operator

in $X=C([0,1],\mathbb{R})$ with the norm $\|f\|_2=\sqrt{\int_0^1 f^2(x)dx}$ we define $T:X\rightarrow X$ by $Tf=gf$ for $g\in X$ How to prove that $T$ is continuous and how to find $\|T\|$ ? I find ...
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Is $T(\ell^1 ) \subseteq \ell^1$?

If we have a linear operator $(\ell^{\infty} , \|\cdot \|_{\infty} ) \rightarrow (\ell^{\infty} , \|\cdot \|_{\infty} ) $ by $T((a_k)_{k \ge 1}) = (b_k)_{k \ge 1}$ where $$ b_k= \frac{a_1+...+a_k}{k}...
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1answer
43 views

Cauchy but not rapidly Cauchy

I want to show that the sequence $\{\frac{(-1)^n}{n}\}$ is Cauchy but not rapidly Cauchy. Here is the work I done so far. I am curiously if I made any errors. Consider the normed linear space $\...
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Normed space, inner product space, which is larger?

My textbook says normed space is inner product space unless it satisfies some parallelogram identity. In this case, we can conclude inner product space is a subset of normed space. However, norm is ...
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A not so obvious corollary?

Let $X$ be a normed vector space over $\mathbb{K}$. Then there is only one completion of $X$, the banach space $\hat{X}$ such that $X$ is a dense subspace of $\hat{X}$. I am trying to prove that there ...
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78 views

How do I link dimension of a normed vector space with closedness?

Let $X$ be a Normed Vector Space, for any $x\in X$ and $r>0$. Let $W:=\{y\in X : \|y-x\|\leq r\}$ and $S:=\{y\in X : \|y-x\|<r\}$ Prove: $W$ is closed if $\dim(X)<\infty$ I can't think of a ...
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31 views

Show two norms are equivalent

Let $$N(z)=\Bigg(\sum_{n=1}^{\infty} \bigg|\frac{c_n}{n}\bigg|^3\Bigg)^{1/3}$$ be a norm on $\ell^3$ where $z=(c_n)_{n\geq1} \in \ell^3.$ Are the norms $N(\cdot)$ and $\|\cdot\|_3$ equivalent? I ...
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14 views

Normed-Spaces and Integrals Question

Notations: $[f]$ is the equivalence class of $f$. $^\ast\int_{\mathbb{R}^n}f$ is the upper integral of $f$ $_\ast\int_{\mathbb{R}^n}f$ is the lower integral of $f$ Functionals $[f]\...
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53 views

Every normed space has a completion?

So I know that a completion of $X$ is a Banach space $Y$ such that $X$ is isometrically isomorphic to a dense subset of $Y$, say $A$. So we need to prove that we can always find a $T \in L(X,A)$ such ...
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How to show that a function in $L_2$ space is uniformly bounded in the “truncated norm” [closed]

Let $$L_2[0, \infty) = \left\{f: \mathbb{R}_{+} \to \mathbb{R}^n \mid \int_0^\infty f(t)^Tf(t) \, dt< \infty \right\}$$ Define the truncated norm as $$\|f\|_K = \sqrt{\int_0^K f(t)^Tf(t) \, dt}$$...
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Expression related to dual norm on bounded linear functionals

Given a vector space $V$ and a norm $\|\cdot\|$ on $V$, the dual norm $\|\cdot\|^*$ on $V^*$ is given by $\|f\|^* = \sup \left\{\frac{f(v)}{\|v\|}\right\}$ over all nonzero vectors $v$. I've found ...
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30 views

Equivalence of two norms.

Suppose, that we have $||.||_1$ and $||.||_2$ norms defined on an arbitrary $X$ vector space. $X$ is a complete space with both of them. For all $x_n, n \in \mathbb{N} \subset X$ series, if $\lim_{n ...
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50 views

For a normal operator is it true that $\|T^*T^2\| = \|T^3\|$?

For a normal operator is it always true that $\|T^*T^2\| = \|T^3\|$? See the accepted answer for the case in a Hilbert space Update: how about $\|T^*T^2\| = \|T\|^3$ in a Hilbert space
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Can we detect smoothness of a norm by its behavior along paths?

We say a norm $\| \cdot \|$ on $\mathbb{R}^n$ is smooth if it is smooth as a function $\mathbb{R}^n\setminus \{0\} \to \mathbb{R}$. (i.e, after restricting the domain). We say a norm is smooth along ...
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Norm vs A-norm in non-Archimedean Functional Analysis

Let $K =(K,| \cdot |)$ be a non-Archimedean valued field. Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that: $||x||=0$ if and only if $x=0$, $||\lambda x||=...
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33 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 ...
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1answer
20 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. ...
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31 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, ...
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29 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 ...