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.
3
votes
3answers
88 views
Maximal Value of Integral
Calculate the maximal value of $\int_{-1}^1g(x)x^3 \, \mathrm{d}x$, where $g$ is subject to the conditions
$\int_{-1}^1g(x)\, \mathrm{d}x = 0;\;\;\;\;\;\;\;$ $\int_{-1}^1g(x)x^2\, \mathrm{d}x = ...
12
votes
5answers
238 views
Passing from induction to $\infty$
Somehow, the operation of passing to the limit after I have shown that something is true by induction for each natural number $n$ troubles me each time. I know there are instances where one cannot ...
3
votes
3answers
78 views
$\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$ for functions in $C([0,1])$?
Why does the following hold for continuous functions on $[0,1]$?
$\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$
0
votes
0answers
45 views
existance of the interpolation space
Let $X\subset L_1+L_2$ and let $Y$ be interpolation space between $L_2$ and $L_{\infty}.$ Given $U:X\longrightarrow Y$. My question is the following:
Is there exists space $Z\subset Y$, such that ...
3
votes
1answer
133 views
$T(V)$ is a closed subspace of $V$?
Let $V$ be a normed vector space (not necessarily a Banach space) and let $S$ and $T$ be continuous linear transformations from $V$ to $V$. If we assume that $T=T \circ S \circ T$. Then how to show ...
2
votes
1answer
98 views
Limit inferior taken on the norm of a sequence
Let $E$ a normed vector space and let $(x_n)$ be a sequence in $E$. Suppose that $x_n$ converges weakly (i.e. wrt the weak topology) to $x$.
Why is it that from the inequality
$$
|f(x_n)| \leq \|f\| ...
0
votes
1answer
50 views
Finding a vector in a n.l.s.
Let $X$ be a normed linear space and $Y$ a closed proper subspace. Prove that for all $\varepsilon > 0$, there is an $x \in X$ with $\|x\| = 1$ and such that $\|x − y\| ≥ 1 − \varepsilon$ for all ...
2
votes
1answer
129 views
Norm closure of convex hull of its set of extreme points
How to prove that the set of extreme points of $B_{\ell^1} = \{v \in \ell^1 : \| v \| \le 1\}$ is $\{ +e^N, -e^N : N=1,2,3,\ldots \}$, where $e^N$ denotes the Nth standard basis element in $\ell_1$: ...
7
votes
2answers
665 views
$T$ is continuous if and only if $\ker T$ is closed
Let $X,Y$ be normed linear spaces. Let $T: X\to Y$ be linear. If $X$ is finite dimensional, show
that $T$ is continuous. If $Y$ is finite dimensional, show that $T$ is continuous if and only if $\ker ...
2
votes
2answers
557 views
Prove that $X'$ is a Banach space
I'm taking a new course on functional analysis and meet with the following problem.
If $X$ is a normed space (not necessarily complete), then prove that $X'$ is a Banach space.
Definition: When the ...
2
votes
0answers
70 views
Consequence of metrizability proof - disregard, the question is an error
In Marian Fabian et al's Functional Analysis and Infinite-Dimensional Geometry, Proposition 3.22 states/proves that if $X$ is a separable Banach space, then the (closed) unit ball, $B_{X^{*}}$ of ...
1
vote
4answers
223 views
Hahn-Banach theorem (second geometric form)
I found in Brezis' Analyse Fonctionelle the Hahn-Banach theorem ("second geometric form") and there is a passage I can't understand. In newest versions of this book the proof has been modified.
...
4
votes
1answer
254 views
Norm for continuous linear functionals, newbie questions
Let $E$ be a normed vector space and let $f\colon E \to \mathbb{R}$ be a continuous linear functional. Define the dual norm of $f$ as
$$
\|f\| = \sup_{\|x\|\leq 1} |f(x)|.
$$
First question. I ...
1
vote
0answers
48 views
Disjointness of stars in a simplicial complex in $\ell_2$
Definitions
Let's consider a full $n$-dimensional simplicial complex $C$ in $\ell_2(X)$. By that I mean a set of all functions $f:X \to[0,1]$ such that $\sum_{x\in X}f(x)=1$ and there are at most ...
2
votes
1answer
331 views
Every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$?
We have just come across the lemma that every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$. My question is that do all norms on $\mathbb{R}^n$ take the form $\|\cdot \|_1, \|\cdot \|_2, ...
2
votes
0answers
107 views
norms on a vector space - is there a quicker way to approach this problem?
I'm looking at the following problem, which I have done but wonder whether there isn't a faster way of doing it. (It's a past exam question which is supposed to take 7.5 minutes, but I only managed to ...
4
votes
1answer
169 views
There exists an isometric embedding
Let $W$ be a closed linear subspace of a normed vector space $V$. Let $i_V: V \to V^{**}$. and $i_W: W \to W^{**}$ be the canonical embeddings of V and W into their second duals. Prove that there ...
-1
votes
1answer
162 views
strictly convex space ---> strictly convex function
How would you prove that in a strictly convex normed vector space, the function $f(x) = \| x \|^2$ is strictly convex??
FYI:
$E$ is strictly convex iff $\| t x + (1-t) y \| <1$ for all $x,y \in ...
2
votes
2answers
238 views
Coercivity vs boundedness of operator
The definition of coercivity and boundedness of a linear operator $L$ between two $B$ spaces looks similar: $\lVert Lx\lVert\geq M_1\lVert x\rVert$ and $\lVert Lx\rVert\leq M_2\lVert x\rVert$ for some ...
1
vote
0answers
249 views
Convergence of $L^p$ norms
Given a measure space $X$ with its measure $\mu$, it can be shown (I'll provide a proof if asked for) that
$\displaystyle \forall f \in L^\infty(X,\mu),~\textrm{if } \exists p_0:\forall q \geq p_0, ...
1
vote
1answer
380 views
A normed space is locally compact iff its closed unit ball is compact.
To prove that A normed space is locally compact if and only if its finite dimensional, I need to prove a
lemma:
A normed space is locally compact if and only if its closed unit ball is compact.
One ...
3
votes
2answers
208 views
Homogeneous function in bounded mean oscillation BMO($\mathbb R^n$) space
Let me recall some notations: The mean oscillation of a locally integrable function $u$ (i.e. a function belonging to $ L^1_{\textrm{loc}}(\mathbb{R}^n))$ over a cube Q in $\mathbb R^n$ (which has ...
2
votes
2answers
74 views
Mean value of convergent series
Let us in a normed linear space have a sequence $\{a_i\}_{i=1}^\infty$ which converges to some value $b$, how can I show that
$$\lim_{n\rightarrow\infty}\sum_{i=1}^n\frac{a_i}{n}=b$$
My idea is to use ...
2
votes
1answer
184 views
Hahn-Banach. Extend the functional by continuity
Let $E$ be a dense linear subspace of a normed vector space $X$,
and let $Y$ be a Banach space. Suppose $T_{0}\in\mathcal{L}(E,Y)$
is a bounded linear operator from $E$ to $Y$. Show that $T_{0}$
can ...
2
votes
2answers
76 views
Showing a function is not continuous in the one-norm
I have the following question:
Consider the normed space $(C([0,1]),||\cdot||_{\infty})$. Let $x_{0}\in [0,1]$ and define $F:C([0,1])\to \mathbb{R}$ by
$F(f)=f(x_{0})$
Show that $F$ is ...
2
votes
1answer
133 views
Continuity with normed spaces
Consider the normed space $(C([0,1]),||\cdot||_{\infty})$. Define $F:C([0,1])\to C([0,1])$ by
$F(f)=f^2$
Show that $F$ is continuous with respect to $||\cdot||_{\infty}$.
I've attempted ...
2
votes
1answer
204 views
How to show convergence in a metric space?
Suppose that $\{x_n\}→x$ where $\{x_n\}$ is a sequence in a normed space V and $x ∈ V$. Show that $\forall y ∈ V, \{x_n + y\} → x + y$.
2
votes
1answer
63 views
Question about normed spaces
Let $(X,||\cdot||)$ be a complete normed space. Let $F_1, F_2, F_3,\ldots\subseteq X$ be closed, non-empty subsets of $X$.
Assume that $F_1 \supseteq F_2\supseteq F_3\supseteq \cdots$
and ...
2
votes
2answers
96 views
Complete normed spaces
Let $(X, ||\cdot||)$ be a complete normed space. Let $||\cdot||$ be a norm on $X$, and assume that there are constants $c_{1}$, $c_{2} \in (0,\infty)$ such that:
$c_{1}||x-y||\le||x-y||_{0}\le ...
0
votes
2answers
95 views
Problem in normed spaces
Some help with the following would be great.
Let $(X,||\cdot||)$ be a normed space.
Let $(x_{n})_{n}$ and $(y_{n})_{n}$ be Cauchy sequences in $(X, D)$. Say also that $s_{n} = ||x_{n} + ...
4
votes
2answers
156 views
“The two notions of boundedness coincide for locally convex spaces”
From Wiki
The boundedness condition for linear operators on normed spaces can be
restated. An operator is bounded if it takes every bounded set to a
bounded set, and here is meant the more ...
37
votes
1answer
867 views
Banach Spaces - How can $B,B',B'', B''', B'''',B''''',\ldots$ behave?
(ZFC)
Let $ \big\langle B,+,\cdot, \:\: \|\cdot\| \:\: \big\rangle $ be a Banach space.
Define $ \mathbf{B} \; = \;\big\langle B,+,\cdot, \:\: \|\cdot\| \:\: \big\rangle $.
Define $\: \mathbf{B}_0 ...
4
votes
1answer
1k views
“Every linear mapping on a finite dimensional space is continuous”
From Wiki
Every linear function on a finite-dimensional space is continuous.
I was wondering what the domain and codomain of such linear function are?
Are they any two topological vector ...
1
vote
1answer
132 views
properties on normed vector space
Let $X \neq \{0\}$ a normed vector space.Prove the following
(a) $X$ does not have isolated points.
(b) If $x,y \in X$ such that $ ||x-y||= \epsilon >0$ then
1.Exists a sequence $(y_n)_n$ in $X$ ...
4
votes
2answers
205 views
How convergence relates to equivalence of norms
Let $X$ be a normed linear space with two norms $||\cdot||_1$ and $||\cdot||_2$.
Prove or disprove that this statements are equivalent:
$||\cdot||_1$ and $||\cdot||_2$ are equivalent,
$\{x_n\}$ ...
3
votes
1answer
134 views
Notation: $L_p$ vs $\ell_p$
$L_p$ is often used to describe a norm, or a vector space with that norm (see e.g. wikipedia).
Is $\ell_p$ (typically, or canonically) a different notation for the same concept, or is it used to ...
2
votes
1answer
106 views
Thinking about open balls in $\mathbb{R}^n$
I've been thinking about open balls in $\mathbb{R}^n$ and whether the density of $\mathbb{Q}$ in $\mathbb{R}$ means that we can find open balls in $\mathbb{Q}^n$ to 'nest inside open balls in ...
4
votes
0answers
143 views
Metric on the unit cube
Let $X$ be $\mathbb{R}^3$ with the sup norm $\|\cdot\|_{\infty}$. Let $Y=\{x\in X: \|x\|_{\infty}=1\}$. For $x,y\in Y,y\neq -x$ define $d(x,y)$ to be the arc length of the path $$Y\cap \{\lambda ...
2
votes
3answers
428 views
Cauchy-Schwarz Inequality
In Luenberger book Cauchy-Schwarz Inequality is defined like this: For all $x,y$ in an inner product space $|(x|y)| \le \|x\|\|y\|$. Equality holds if and only if $x = \lambda y$ or $y = \theta$.
...
14
votes
3answers
522 views
Intersection between orthogonal complement of a subspace and a set
Consider the normed vector space $E=\mathbb{R}^n$. Define
$ P=\{x \in \mathbb{R}^n: x_i \geq 0, \forall i \}$.
Let $M$ be a subspace such that $M \cap P = \{0\}$. I want to see that $M^\perp \cap ...
0
votes
2answers
211 views
Convergence in the uniform norm topology, convergence wrt the uniform norm and uniform convergence
From Wikipedia
Given a topological space $X$, we can equip the space of bounded real
or complex-valued functions over $X$ with the uniform norm topology.
Then uniform convergence simply means ...
5
votes
0answers
187 views
Are Arzelà–Ascoli theorems results of similar theorems on normed spaces, metric spaces or other spaces?
From Wikipedia, two generalizations of the Arzelà–Ascoli theorem are
Let $X$ be a compact Hausdorff space. Then a subset $F$ of $C(X)$,
the set of real-valued continuous functions on X, is ...
1
vote
1answer
147 views
Problem involving a hyperplane and affine subspace II
I am trying to solve this little problem.
Suppose you have a normed vector space $E$. Let $H$ be a hyperplane ( $H=\{x\in E: f(x)= \alpha\}$ for some linear functional $f$ and some real number ...
2
votes
1answer
52 views
P-adic “Norm” and scalability criterion
I just came across the p-adic norm for the first time. I tried to show that it is actually a norm on $Q$ but I was asking myself, whether checking scalability is a bit self referential ?
What I mean ...
3
votes
2answers
183 views
A problem involving a hyperplane and an affine subspace
Any ideas on how to solve this??
Let $E$ be a normed vector space and let $H \subset E $ be a hyperplane. Let $V \subset E$ be an affine subspace containing $H$. Prove that either $V=H$ or $V=E$.
If ...
2
votes
0answers
57 views
Embedding tree metric isometrically into $\ell_\infty$
I just started (independent) learning on metric embeddings from the Fall 2003 offering of the course at CMU. I have a limited mathematical background and alas, it made me stumble at the first exercise ...
2
votes
0answers
106 views
Different topologies on a normed/inner product space
Given a normed vector space $X$. Let $\mathcal{T}$ represent the topology on $X$ induced by the norm.
Define:
$A$:={topologies that can make the norm continuous},
$B$:={topologies that can make the ...
3
votes
2answers
1k views
Is a norm a continuous function?
Is a norm on a set a continuous function with respect to the topology induced by the norm?
Is a topology on the set that can make the norm continuous (i.e. the topology that is compatible with the ...
2
votes
1answer
120 views
Compute an operator norm
Consider the operator $M$ acting on the space $\mathbb{R}[X]$ of real polynomials by $Mp(x)=xp(x)$. We equip $\mathbb R[X]$ with the $L^2$ norm
$$
\|p\|^2=\int p(x)^2d\mu(x),
$$
where $\mu$ is a ...
1
vote
1answer
164 views
To construct a counterexample of normed space
Please construct a counterexample for the following: $A$ is normed space and $M$ is a dense subspace of $A$, if there is a functional $f$ such that $f(M) = 0$, then $f=0$.
Besides, if $A$ is a Banach ...
