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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55 views

Let $X=\left\{ f:[0,1]\to\mathbb R \;\big|\; \lVert f\rVert_X<\infty,\ f(0)=0\right\}$. Show $(X,\lVert\cdot\rVert_X)$ is complete.

The following is a problem on an old Analysis preliminary exam at my institution; I'm prepping for the prelim. The problem is: Let $$X=\left\{ f:[0,1]\to\mathbb R \;\big|\; \lVert f\rVert_X<\infty,...
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1answer
79 views

Show that the following is a bounded linear operator on $L^2(R_+)$ Calculate the adjoint operator.

The following is a question I have been working on for some time with help from my teacher. Unfortunately we have a solution but are not 100% confident with it. Some guidance on if part(s) of our ...
1
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0answers
49 views

Showing $\|x+y\|^2+\|x-y\|^2=2\|x\|^2+2\|y\|^2 \Rightarrow \| \cdot \| $ is induced by scalar product

I need to show the above $\forall x,y,v \in V$ , a normed vector space on $\Bbb R$. A hint was given that i should first show that $$s:V \times V \to \Bbb R ; \: \:\: s(u,v):=\frac1 4 (\|u+v\|^2-\|u-v\...
2
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1answer
31 views

If $A$ is bounded and dissipative, is $\lambda \mathbb 1-A$ invertible for $\lambda>0$?

Let $X$ be a Banach space and $j$ a map (not necessarily linear or anti-linear!) from $X$ to $X'$ so that $$j_x(x)=\|x\|^2 \qquad \forall x \in X \text{ and }\qquad \|j_x\|_{X'}=\|x\|$$ We say that ...
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0answers
21 views

Entrywise expression for L2 matrix norm

The matrix norm induced by the $\ell^2$ norm is known to be equal to the maximum singular value of the matrix. The matrix norms induced by the $\ell^1$ and $\ell^\infty$ norms admit simple ...
2
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1answer
25 views

The quotient norm on $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$.

I try to show that the norm on the quotient space $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$, where $x = (x_n)_{n \in \mathbb{N}} \in \ell^{\infty} (\...
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0answers
28 views

Together with the algebra of cardinal numbers, is there analysis of cardinal numbers? [closed]

Let $C$ be the collection of all cardinal numbers. Is there any norm, inner-product, metric (other than discrete metric), topology(other than discrete, co-finite topology) on $C$, which is very useful?...
3
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0answers
39 views

Closed map $T:X \to Y$ has closed graph?

Let $T:X\to Y$ be a linear operator between two normed vector spaces. My question is: If $T$ is a closed map (sends closed sets to closed), then is the graph of $T$ a closed set of $X \times Y$? ...
0
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1answer
37 views

Equivalence of statements about a linear map

I need someone to help me solve the following exercise: Let $(X, ||\cdot||_X)$ and $(Y, ||\cdot||_Y)$ be normed vector spaces over a common field $\mathbb K$ $(\mathbb R$ or $\mathbb C)$. For a ...
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1answer
42 views

Exercise 1.65 of Megginson's “An Introduction to Banach Space Theory”.

Unfortunately I do not succeed in completing the following exercise: Let $X$ be a Banach space and let $T : X \to \ell^{1} (\mathbb{N})$ be a linear operator. For each $n \in \mathbb{N}$, let $(Tx)...
2
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1answer
26 views

Weak/Weak* topologies compared to topologies generated by semi-norms from dense subset

The weak topology on normed linear space $X$ can be defined as being induced by semi-norms $\|\cdot\|_{x'}$, $x'\in X'$ with $\|x\|_{x'}=|x'(x)|$. Similarly the weak* topology is induced by $\|\cdot\|...
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0answers
34 views

The precise definition of Cartesian coordinate and Euclidean space?

I'd searched them for a while, but still have not found a clear and unity definition on it. The problem really confused me. What is the precise definition of Cartesian coordinate and Euclidean space? ...
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0answers
17 views

Conditions for non-normability of (nontrivial proper and nondense) subspace of non-normable space.

Lets $X$ be a non-normable topological vector space and let $Y\subset X$ be a proper subspace. Clearly if $Y$ is dense in $X$ then $Y$ must be non-normable too. Can we have that conclusion with weaker ...
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1answer
30 views

Proof/disprove contunuity of a map [duplicate]

I need help with proving / disproving something: Look at the map $$\Phi: (C([0,1], \mathbb R), ||\cdot||_{\infty}) \to (\mathbb R, |\cdot|); \,\,\,\,\,\,\Phi(u) := \int_0^1 u²(t) dt$$ a) ...
1
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1answer
37 views

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 ...
2
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0answers
45 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 ...
3
votes
1answer
34 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 ...
0
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2answers
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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2answers
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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1answer
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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1answer
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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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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1answer
59 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
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 ...
3
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1answer
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}(\...
2
votes
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 ...
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 ...
0
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1answer
32 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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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 ...
2
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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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2answers
46 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=...
0
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1answer
22 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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1answer
17 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 ...
0
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1answer
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 ...
0
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1answer
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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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 $B_\epsilon (\alpha x)\subset E$. Can ...
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1answer
50 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 ...
0
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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
10 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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1answer
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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1answer
30 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,...
2
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1answer
30 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
18 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
18 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
53 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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votes
2answers
27 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= \frac{a_1+...+a_k}{k}...
1
vote
1answer
47 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 $\...
0
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0answers
25 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 ...
0
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0answers
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 ...