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

Unit Ball in 1 norm is open in ($C[0,1] , || \quad ||_{\infty}$)

Claim $B_1(0,1) := \{ f \in C[0,1] ; ||f||_{1} < 1 \} $ in $(C[0,1],||\quad || _1)$ is open in $(C[0,1],||\quad || _{\infty}).$ We need to take any $f \in B_1(0,1),$ and we have to find an ...
2
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0answers
129 views

Completely metrizable space, complete norm

Let $E$ be a normed, completely metrizable space. Prove that the initial norm is complete. How can I go about solving this problem? I will be grateful for all your hints. Thank you!
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2answers
79 views

Norm equivalence (Frobenius and infinity)

I am trying to prove the matrix norm equivalence for norms 1, 2, $\infty$ and Frobenius. I have managed to solve find the constants for $||.||_{1}$ and $||.||_{2}$ but I cannot see how to continue ...
2
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1answer
38 views

How to compute the induced matrix norm $\| \cdot \|_{2,\infty}$

The induced norm of the matrix $A$ as a map from $(\mathbb R^n , \| \cdot \|_p)$ to $(\mathbb R^n, \| \cdot \|_q)$ is given by $$ \| A \|_{p,q} = \sup_{x\in\mathbb{R}^n\setminus \{0\}} ...
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53 views

Lipschitz continuity and gradient of a real-valued function on a normed space

The gradient of a function $f:E\to\mathbb{R}$ is Lipschitz continuous with parameter $L > 0$ iff $$\|\nabla f(x) - \nabla f(y)\|^* \le L\|x-y\| \quad \forall x,y\in E.$$ I have two questions: ...
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40 views

Show that a subspace of a normed vector space is closed

Let $X$ be a normed vector space over $\mathbb K, \mathbb K = \mathbb R$ or $\mathbb K=\mathbb C.$ Let $Y$ be a closed linear subspace of $X$ and $x\in X\backslash Y.$ Set $Z=\{y+\alpha x;\;y\in ...
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1answer
49 views

Computing the norm of a linear operator

For two finite-dimensional real vector spaces $E_1,E_2$, define an linear operator $A:E_1\to E_2^*$. Its adjoint operator is defined by $A^*:E_2\to E_1^*$ its adjoint operator, i.e. $$\langle Ax,u ...
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2answers
31 views

$C^n[a,b]$ as a normed algebra

I would like to prove that the space of complex valued functions, differentiable $n$ times with continuous derivative, $C^n[a,b]$, with the metric defined by the norm ...
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1answer
30 views

When Heine - Borel theorem holds

If $\cal A$ is an abelian Banach algebra, then its maximal ideal space $\Omega$ is a compact Hausdorff space. In the proof of this theorem, the author says, since $\Omega \subset ball \cal A^*$, it ...
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44 views

Every homomorphism on a C*-algebra is a *-homomorphism

The following is a proposition of Conway's Functional analysis: and also he uses below exercise to proof above proposition: But I do not know how he uses the Exercise and say $||h||=1$. Please ...
3
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98 views

An open set in the space of bounded real sequences

Let $X$ denote the set of all bounded real sequences, equipped with the norm $\| (x_n)\|_\infty:= \sup\{|x_1|,|x_2|,|x_3|,\ldots\}$; Let $X_{++}$ denote the set of all bounded positive real sequences ...
2
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1answer
50 views

Example of a proper, dense ideal in a unital Banach algebra

I know that if ${\cal A}$ is a non- unital Banach algebra, then ${\cal A}$ may contain some proper, dense ideals. I need an example of a proper, dense ideal in a unital Banach algebra or show that it ...
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1answer
26 views

Extend Metric Space Challenge

Let $(E, D)$ be a metric space. Consider $D_1: E\times E \to \mathbb{R}$ where $$ D_1(x,y)=\frac{D(x,y)}{1+ D(x,y)}. $$ I read some note about it but I want to find why $D_1$ is also a metric and ...
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1answer
53 views

Show that $\infty$-norm and $C^1$-norm are not equivalent.

Show that $\infty$-norm and $C^1$-norm are not equivalent. For the $C^1([a,b],\mathbb{R})$ space, show that $\displaystyle ||g||_\infty=sup_{a\leq t\leq b}|g(t)|$ and $\displaystyle ...
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2answers
23 views

Are lines in arbitrary normed vector spaces closed?

Let $(V, \| \cdot \|)$ be a normed (real) vector space. Given two vectors $a$ and $d$ (with $d$ not the zero vector), is the line $ L = \{a + td: t \in \mathbb{R}\} $ through $a$ in direction $d$ ...
3
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2answers
506 views

Examples of compact sets that are infinite dimensional and not bounded

In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, ...
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43 views

Convergence of $a_n=d(u_n,\Bbb{Z})$ where $u_n=\bigl(\frac{1+\sqrt{5}}{2}\bigr)^n$.

I would like to study the sequence defined by $$ a_n=d(u_n,\Bbb{Z}). $$ Where $u_n=\bigl(\frac{1+\sqrt{5}}{2}\bigr)^n$ and $d(u_n,\Bbb{Z})=\inf\{d(u_n,x):x\in\Bbb{Z}\}$. I do not really ...
2
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1answer
43 views

Quotient of a Banach space $X$ gets quotient topology under standard norm induced from $X$.

Let $X$ be a Banach space and $Y$ be a closed subspace of $X$. Then there is a canonical norm on $X/Y$. I want to show that this norm induces quotient topology on $X/Y$. Any hint/solution? I was ...
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1answer
33 views

Prove that for $0<p<1$, $|x-y|^p$ is a metric space on $\mathbb R^{n}$

Define the function $f_p : \mathbb R^{n} \to \mathbb R^{n}$ for $n ≥ 2$ by $f_p(x) = \sum_{k=1}^{n} \lvert x\rvert^{p}$. Show that for $0<p<1$, we get $d_b(x,y) = f_p(x-y)$ is a metric on ...
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1answer
29 views

Show that an operator is weakly compact

If $(X,\Omega,\mu)$ is a finite measure space, $k\in L^\infty(X\times X, \Omega\times \Omega,\mu \times \mu)$ , and $K:L^1(\mu)\to L^1(\mu)$ is defined by $$(Kf)(x)=\int k(x,y) f(y) d\mu(y)$$ show ...
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0answers
52 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 ...
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1answer
52 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 ...
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1answer
30 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 $ran \phi^*$ ...
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0answers
36 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 ...
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2answers
48 views

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

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
33 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 ...
3
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1answer
64 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
39 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 ...
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0answers
34 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
28 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
57 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
47 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 ...
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2answers
29 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
25 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 ...
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1answer
46 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 ...
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1answer
65 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
64 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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46 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 ...
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1answer
41 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 ...
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1answer
47 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
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1answer
26 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
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1answer
74 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 ...
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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 ...
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1answer
35 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 ...
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1answer
52 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 ...
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1answer
36 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
97 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 ...
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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 ...
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1answer
81 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!
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1answer
78 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 ...