Tagged Questions

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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41 views

Weakly compact operator on $c_0$ is compact

Show that if $T\in {\cal B}(c_0)$ and $T$ is weakly compact, then $T$ is compact. My attempt: $T$ is weakly compact, so there is a reflexive space $X$ , and operators $A\in {\cal B}(X,c_0) $ and $B ...
2
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1answer
41 views

If any two norms on a vector space are equivalent then the space is finite-dimensional [duplicate]

I need to prove: If any two norms on a vector space are equivalent then the space is finite-dimensional. I am aware of the converse of this result that on a finite dimensional vector space any two ...
0
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1answer
18 views

Dual operator of an isometry

If $X,Y$ are Banach spaces and $\phi:X\to Y$ is an isometry, show that $\phi^*$ is surjective. I can use the equality $^\perp(ran \phi^*) = \ker\phi=\{0\}$, and also use the fact that $ran \phi^*$ ...
1
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0answers
34 views

Extending linear continuous functions.

Let $E$, $F$ be normed vector spaces and $M$ a subspace of $E$. I'm trying to find an example of a function $f:M\to F$ such that $f$ is linear and continuous but that you can't extend it to ...
0
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2answers
35 views

Show that if X is a normed linear space, then any finite-dimensional subspace M of X must be closed.

It suffices to show that any proper subspace M of X is closed, since if M is not proper the result is trivial. I am unsure how to approach this proof. Contradiction seems a little messy, as supposing ...
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0answers
30 views

Finite dimensional spaces and R^n

I have a couple of questions, any assistance would be appreciated. I know that it can be shown that any finite dimensional space $M$ of dimension $N < \infty$ endowed with an inner product can be ...
2
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1answer
49 views

show that every compact operator on Banach space is a norm-limit of finite rank operators (in a particular way, under the given hypotheses)

Let $X$ be a Banach space and suppose there is a net $\{F_i\}$ of finite-rank operators on $X$ such that $(a)$ $\sup_i||F_i||<\infty$ ; $(b)$ $||F_ix-x||\to 0$ for all $x$ in $X$. Show that if $A$ ...
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0answers
32 views

Inner product spaces, normed spaces, metric spaces and topological spaces

I am collecting theorems or properties that hold in IPS, NS, MS or topological spaces, but not all of them. The reason is that I want to create some sort of overview over the respective spaces and ...
1
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0answers
18 views

An example of a non-compact operator which is equal to (norm) limit of compact operators

For a normed linear space $X$ and a Banach space $Y$, the set of all compact operator from $X$ to $Y$, which is denoted by $K(X,Y)$, is normed closed in $B(X,Y)$. Is there a counterexample which ...
2
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0answers
27 views

Maximal chain in the collection of all invariant subspaces for compact operator $K$

Let $X$ be a Banach space over ${\Bbb C}$, and $K\in K(X)$ ($K(X) = $ compact operators space). Show that if ${\cal L}$ is a maximal chain in the $Lat K$ ($Lat K = $ the collection of all invariant ...
2
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1answer
54 views

Operators on non-separable Banach spaces have non-trivial invariant subspaces

Show that if $T\in B(X)$ and $X$ is not separable, then $T$ has a nontrivial invariant subspace. I know that $\ker (T)$ and $\operatorname{ran}(T)$ are invariant $T$-subspace. So if $\ker T\neq ...
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0answers
37 views

functional analysis: show L^1 integral operator has norm 1

I just started my course in functional analysis and have already stumbled across some things I don't understand, which are quite basic :(. In my lecture notes it says: Let $\mu$ be a measure on a ...
1
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2answers
26 views

Prove that $V$ is a complete space

Let $V$ be the space of real sequences with a finite number of elements $\neq 0$ ($ \exists N_x$ so that $x_k=0 \forall k>N_x$. Define $$||\vec x||_1=\sum_{k=1}^{N_x}|x_k|$$ Prove that $V$ is a ...
2
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0answers
24 views

Invariant subspaces and their complements

The following is Exercise 6.4.2 of Conway's Functional Analysis: Suppose $T\in B(X)$ where $X$ is a Banach space. Prove that $M$ is invariant under $T$ if and only if $M^{\perp}$ is invariant under ...
2
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1answer
45 views

Compactness of an operator on $c_0$ in terms of its infinite matrix representation

Let $A\in {\cal B}(c_0)$ (${\cal B}(c_0)$ is linear bounded operators on $c_0$) and for $n\geq 1$, define $e_n \in c_0$ by $e_n(m)=\delta_{nm}$. Put $\alpha_{nm}=(Ae_n)(m)$ for $n,m\geq 1$. we have $M ...
1
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1answer
53 views

$T\in\mathcal{L}(X,Y)$ maps closed bounded subsets onto closed subsets $\implies$ Range $T$ is closed.

Given two normed spaces $X$ and $Y$ and let $T$ be a bounded linear operator $T:X\to Y$. Assume that $T$ maps bounded and closed subsets of $X$ onto closed subsets of $Y$. Show that the range of $T$ ...
3
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1answer
54 views

Give necessary and sufficient conditions for a multiplication on $L^p$ to be compact

Let $(X, \Omega, \mu)$ be a $\sigma-$ finite measure space and for $\phi \in L^\infty(\mu)$ let $M_\phi:L^p(\mu) \to L^p(\mu)$ defined by $M_\phi f = \phi f $ be the multiplication operator. Give ...
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0answers
41 views

Is every subspace of a normed linear space which is not closed a hyperspace.

Let $B \subset X$ where $X$ is a normed linear space over $\mathbb{R}$ and $B$ is a proper subspace. If $B$ is not closed, is $B$ necessarily a hyperspace(maximal proper subspace) in $X$. I attempted ...
0
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1answer
40 views

Find the coefficients $a,b$ so that $\|\vec x\|=a(|x_1|+|x_2|)+b\max(|x_1|,|x_2|)$ is a norm

For which coefficients $a,b$, the expression: $$\|\vec x\|=a(|x_1|+|x_2|)+b\max(|x_1|,|x_2|)$$ is a norm in $\mathbb R^2$? My attempt: I need to verify the properties of the norm: Triangle ...
0
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1answer
41 views

Identity mapping from $C^1[0,1]$ with supnorm to $C^1[0,1]$ with $C^1$ norm is not continuous

If X = $C^1[0,1]$, $||f||_1 = ||f||_\infty$, and $||f||_2 = ||f||_\infty + ||f'||_\infty$, show that the identity map $I: (X, ||·||1) → (X, ||·||2)$ is bounded below, but not continuous. I know ...
2
votes
1answer
21 views

$\sup$ norm of a function

The following is an example of Murphy's C*-algebras and operator theory: I do not know how he concludes $$\int_0^1 |k(s,t) - k(s',t)||f(t)| dt \leq \sup|k(s,t) - k(s',t)|||f||_\infty$$ Please help ...
3
votes
1answer
56 views

Integral operator on $L^p$ is compact

Let $(X,\Omega,\mu)$ be an arbitrary measure space, $1<p<\infty$ , and $\frac{1}{p}+ \frac{1}{q} = 1$. If $k:X. X\to \Bbb C$ is an $\Omega.\Omega-$ measurable function such that $$M = [\int ...
1
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1answer
16 views

Condition under which a locally convex topological vector space becomes a normed linear space

Is this true that a locally convex topological (Hausdorff) vector space becomes a normed space when its local base has only one element, so only one Minkowski functional and so only one seminorm and ...
1
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1answer
33 views

geometric interpretation of the norm: $\|\vec x\|={(|x_1|+|x_2|)\over 3}+{2\max(|x_1|,|x_2|)\over 3}$

Let $p:\mathbb R^2 \to \mathbb R$ be a norm so that $$\|\vec x\|={(|x_1|+|x_2|)\over 3}+{2\max(|x_1|,|x_2|)\over 3}$$= $${{\|\vec x\|_1\over 3}}+{2\|\vec x\|_\infty\over 3}$$ The thing is that I need ...
1
vote
1answer
43 views

Why does this proof fail?[convergence of infinite sums]

An equivalent way of saying that a normed vector space is complete is saying that every absolutely convergent series, converges. Hence' in some normed vector-space(incomplete), there must be a ...
0
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1answer
33 views

Understanding the scalar product in the duality

I was trying to solve an exercise for my class, but then I have found somewhere a solution. I need to understand the meaning of a certain step. The exercise and the solution read as follow. Exercise ...
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0answers
64 views

Dual norm of the matrix L1 norm is infinity norm (and vice versa)

Recall that for a given norm $\|\cdot\|$ on $\mathbb{R}^n$, the dual norm is defined as a function $\|\cdot\|_*: \mathbb{R}^n \rightarrow \mathbb{R}$ with: $\|y\|_* = \max \limits_x \{x^Ty: \|x\|\le1 ...
1
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1answer
58 views

What kind of space is this: $\Bbb{R}^n\times\Bbb{S}_{++}^n$?

Let $\Bbb{R}^n$ be the Euclidean space of $n$-dimensional column vectors with real coefficients. Moreover, $\Bbb{S}_{++}^n$ be the space of symmetric positive definite $n\times n$ real matrices. We ...
1
vote
1answer
76 views

Is Frobenius norm induced up to a scalar factor?

I know that the Frobenius norm is not induced since $||I||_F=\sqrt n\neq 1$. But what if we consider the norm $\frac 1 {\sqrt n} ||\cdot ||_F$? Thank you!
1
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1answer
70 views

Does one need the Hahn-Banach theorem to prove the mean value inequality for maps into a normed space?

Consider the following mean value theorem: If $f$ is a continuous mapping of $\,[a,b]$ into a normed linear space $X$, whose norm doesn't derive from an inner product, and $f$ is differentiable on ...
4
votes
1answer
69 views

Riesz's Lemma for $l^\infty$ and $\alpha = 1$

Riesz's Lemma says the following: Let $X$ be a normed vector space and $Y$ a proper closed subspace of $X$. Pick $\alpha \in (0,1)$. Then $\exists x\in X$ such that $|x|=1$ and $d(x,y) \geq \alpha$ ...
3
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1answer
63 views

Why is semi-norm special?

One difference between semi-norm and norm is: "It is possible for ||v|| = 0 for nonzero v, ||.|| being semi-norm" I see some papers, and they use semi-norm directly. Why is semi-norm better than ...
5
votes
1answer
95 views

Must a normed vector space be over $\mathbb{R}$ or $\mathbb{C}$?

If it must be, why is this so? In the maths courses I have taken normed vector spaces always have been over $\mathbb{R}$ or $\mathbb{C}$, but I don't see that this has to be so. I am asking because I ...
3
votes
0answers
74 views

Condition on Equality of closure of open ball and closed ball, suppose I have a counterexample

Let $(X, d)$ be a metric space. Also for $x \in X$ and $r \ge 0$ define: $$ B(x,r) = \{ y \in X : d(x,y) < r \} \quad \mbox{ and } \quad K(x,r) = \{ y \in X : d(x,y) \le r \}. $$ Denote by ...
1
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1answer
40 views

Proof of compactness for sets of norm equal to one in finite-dimensional normed vector spaces

The proposition I have been trying to prove is that the set $A=\{x\in E:N(x)=1\}$ is a compact subset of the (real) finite-dimensional vector space $E$ for any norm $N:E\to \mathbb{R}$. I am reading ...
6
votes
1answer
161 views

Radon-Riesz & Kadec-Klee

Let us say that a normed vector space has the a) RR (Radon-Riesz) property if for any sequence, norm convergence is equivalent to weak convergence + convergence of norms. b) KK (Kadec-Klee) property ...
1
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1answer
57 views

Spectrum of a finite rank operator

If $ T\in B(H)$ is a finite rank operator, then there are orthonormal vectors $e_1,...,e_n$ and vectors $g_1,...,g_n$ such that $Th=\sum_{i=1}^n (h,e_i )g_i$, then we can easily see that $T$ is ...
5
votes
1answer
113 views

What is an ultrametric normed vector space?

Wikipedia's article on ultrametric spaces seems to suggest that an ultrametic space can also be a normed vector space. It seems to be impossible for an ultrametric to be induced by a vector space ...
0
votes
2answers
59 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 $T$ ...
2
votes
0answers
30 views

Extension of function with values in a Banach space

I want to prove the following Let $E,X$ be Banach spaces, and $Y\subset E$ a closed subspace with codimension $1$. Let $T:Y \to X$ be a continuous linear function. Then there exists a continuous ...
0
votes
3answers
42 views

In a normed vector space, does $x+r\Gamma = x'+r'\Gamma$ imply $x=x',r=r'$?

Let $X$ denote a real normed vector space and suppose $\Gamma \subseteq X$ is a bounded subset with two or more elements. Consider $r,r' \in \mathbb{R}_{> 0}$ and $x,x' \in X$. Does $x+r\Gamma = ...
3
votes
0answers
29 views

give necessary and sufficient conditions that every functional in $w^*-cl M$ be the $w^*$- limit of a sequence from M

Let $X$ be a separable Banach space. If $M$ is a linear manifold in $X^*$ give necessary and sufficient conditions that every functional in $w^*-\mathrm{cl} M$ be the $w^*$- limit of a sequence from ...
3
votes
1answer
26 views

convex weak* sequentially closed subset of a separable Banach space implies weak* closed

I'm studying Conway's a course in Functional Analysis by myself. The following is corollary 6.12.7 of this book. If $X$ is a separable Banach space and $A$ is a convex subset of $X^*$ that is weak* ...
3
votes
0answers
92 views

How to prove that this set is closed. [duplicate]

Lets say $a_1, a_n$ are normed vectors. Why is the set $C = \{\Sigma_{i=1}^n \lambda_ia_i: \lambda_i \ge0\}$ closed? The $\lambda$'s can be any non-negative numbers. So C is the set of all ...
0
votes
1answer
19 views

In a normed vector space, if $S⊆X$ is closed, is $f(S,r)$ necessarily closed?

Let $X$ denote a (real or complex) normed vector space. Consider the function $f : \mathcal{P}(X) \times \mathbb{R}_{\geq 0} \rightarrow \mathcal{P}(X)$ given as follows. $$f(S,r) = \bigcup_{s \in ...
2
votes
0answers
36 views

Adjoint of an operator on $C(X)$

Let $X,Y$ be compact and $\tau: Y\to X$ be continuous. If $A:C(X)\to C(Y)$ such that $(Af)(y)=f(\tau(y))$ is a linear and continuous operator, then show that the adjoint operator $A^*:M(Y)\to M(X)$ is ...
0
votes
0answers
68 views

Show that an hyperplane is closed iff f is linear and continuous

I need an help with the following exercise. Let $(E,\| \cdot \|)$ a n.v.s. and let $f:E\rightarrow \Bbb R$. Show that $H=\{x\in E: f(x)=\alpha\}$ is closed if and only if $f\in E'.$ Actually, I ...
0
votes
1answer
32 views

Show that an operator is bounded.

Let $\{\alpha_{mn} ;m,n\geq 1\}$ be scalars satisfying a- $M=\sup_n\sum_{m\geq 1}|\alpha_{mn}|<\infty $ , and b- $\sup_n|\alpha_{mn}|<\infty$, then $(Af)(n) = \sum_{m\geq 1}\alpha_{mn} f(m)$ ...
5
votes
1answer
65 views

When is $M+N$ closed

Let $X$ be a Banach space and $M,N$ be closed subspaces. If the range of linear transformation $x\to (x+M)\oplus (x+N)$ from $X$ into $X/M\oplus X/N$ is closed show that $M+N$ is closed. or using ...
0
votes
0answers
16 views

Converse to sequential Banach--Alaoglu [duplicate]

Let $B$ be the closed unit ball of the dual space of a real normed vector space $V$. If $V$ separable then $B$ is sequentially compact in the weak-* topology. What about the converse?