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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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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24 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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1answer
24 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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1answer
34 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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7k views

Why is $L^{\infty}$ not separable?

$l^p (1≤p<{\infty})$ and $L^p (1≤p<∞)$ are separable spaces. What on earth has changed when the value of $p$ turns from a finite number to ${\infty}$? Our teacher gave us some hints that ...
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1answer
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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27 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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35 views

Can I have a map to follow in this proof?

Let X be a normed vector space, $Y:=B(X,\mathbb{K})$ a vector space over $\mathbb{K}$ of the bounded linear functionals with the usual norm $$||f||=sup_{||x||=1, x\in X}|f(x)|, f\in Y$$ and$$F(x) ...
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2answers
427 views

Poincaré inequality using $H^1$ seminorm

Does this inequality holds for Poincaré Inequality? $$||v||_{L^2} \leqslant C_p |v|_{H^1}$$ and $$ |v|_{H^1} = ||v'||_{L^2} $$ where $| \dot~ |$ is the semi norm and $||\dot~||$ is the norm. I'm ...
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1answer
45 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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1answer
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 ...
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1answer
25 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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1answer
145 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: $D(P) = D(Q) = \mathcal{S}(\mathbb{R})$ ...
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1answer
2k views

Proof of separability of Lp spaces

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces: I have a few questions regarding the proof. Questions: It says 'it is easy to construct a function $f_{2} ...
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1answer
299 views

Examples of strictly convex normed spaces that are not uniformly convex

I am studying approximation theory, just to learn basically, and in different books I saw a theorem about uniqueness of best approximation. One book uses strict, the other uses uniform convexity with ...
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1answer
30 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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2answers
31 views

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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0answers
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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1answer
18 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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2answers
43 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 ...
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1answer
15 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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1answer
28 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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333 views

why must a normed space homeomorphic to a complete metric space be complete?

Why must a normed space X homeomorphic to a complete metric space Y be complete? I've solved this in the case where Y is a normed space (considering open balls gives Lipschitz equivalence), but am ...
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1answer
31 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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149 views

Regarding linear independence on a normed-linear space given a condition

Let $(X,\|\cdot\|)$ be a normed linear space and $x_{1}, x_{2}, \ldots, x_{n}$ be $n$ linearly independent vectors in $X$. Show that there exists $\epsilon > 0$ such that if $y_{1}, y_{2}, ...
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0answers
16 views

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 $B_\epsilon (\alpha x)\subset E$. ...
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2answers
103 views

Non-subdifferentiable convex function on a normed space

Is there any convex function $f$ on a normed space $X$ such that $\partial f(x)=\emptyset$ at every $x\in X$?
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43 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
1k views

Why is semi-norm special and preferred?

One difference between semi-norm and norm is: "It is possible for $\|v\| = 0$ for nonzero v, $\|\cdot\|$ being semi-norm" I see some papers, and they use semi-norm directly. Why is semi-norm ...
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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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0answers
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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0answers
18 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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1answer
27 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 ...
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1answer
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 ...
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1answer
26 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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0answers
16 views

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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1answer
16 views

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
51 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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1answer
39 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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26 views

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= ...
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77 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 ...
4
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1answer
793 views

Equality in Young's inequality for convolution

I am trying to understand how sharp Young's inequality for convolution is. The inequality says $||f \ast g||_r \leq ||f||_p ||g||_q$ where as $1/p+1/q = 1+1/r$. Actually, there are a couple of papers ...
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2answers
521 views

continuity in the strong topology implies continuity in the weak one

I have to prove that if $T:(E,\|\cdot\|_E)\rightarrow (F,\|\cdot\|_F)$ is a continuous and linear operator, and $x_h\rightharpoonup x$ in $E$, than $Tx_h\rightharpoonup Tx$ in $F$. So we know that $T$ ...
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0answers
24 views

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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43 views

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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1answer
51 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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5k views

Proof that every finite dimensional normed vector space is complete

Can you read my proof and tell me if it's correct? Thanks. Let $V$ be a vector space over a complete topological field say $\mathbb R$ (or $\mathbb C$) with $\dim(V) = n$, base $e_i$ and norm ...
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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? ...
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11 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 ...
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2answers
23 views

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) \, ...