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.
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 ...
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 ...
3
votes
1answer
453 views
Are isometric normed linear spaces isomorphic?
I should know the answer to this (and I did some time ago, but have forgotten): If the normed linear spaces $X$ and $Y$ are isometric (there is a bijective map from $X$ to $Y$ that preserves ...
8
votes
3answers
334 views
Is there an easy example of a vector space which can not be endowed with the structure of a Banach space
Let $V$ be a real vector space.
Is there always a norm on $V$ such that $V$ is complete with respect to this norm?
If not, is there an easy counterexample?
4
votes
1answer
141 views
Limit of a sequence in a vector normed space
I have this problem I'm working on.
Hints are much appreciated (I don't want complete proof):
In a vector normed space, if $ \{x_n\} \longrightarrow x $ then $ z_n = \frac{x_1 + ...+x_n}{n} ...
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 ...
25
votes
0answers
431 views
Explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$
Do you know an explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$? Using the axiom of choice, every vector space admits a norm but have you an explicit formula on ...
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 ...
4
votes
1answer
204 views
Shortest path on unit sphere under $\|\cdot\|_\infty$
Let $X$ be $\mathbb{R}^3$ with the sup norm $\|\cdot\|_{\infty}$. Let $Y=\{x\in X: \|x\|_{\infty}=1\}$. For $x,y\in Y$ define $d(x,y)$ to be the arc length of shortest paths on $Y$ joining $x,y$. (It ...
4
votes
1answer
204 views
Non-completeness of the space of bounded linear operators
If $X$ and $Y$ are normed spaces I know that the space $B(X,Y)$ of bounded linear functions from $X$ to $Y$, is complete if $Y$ is complete. Is there an example of a pair of normed spaces $X,Y$ s.t. ...
1
vote
3answers
333 views
Interior of a Subspace
There is a conjecture: "The only subspace of a normed vector space $V$ that has a non-empty interior, is $V$ itself." (here, the topology is the obvious set of all open sets generated by the metric ...
12
votes
2answers
2k views
Difference between metric and norm made concrete: The case of Euclid
This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me.
This time I am making ...
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 ...
13
votes
1answer
388 views
Multiplicative norm on $\mathbb{R}[X]$.
How to prove that : there is no function $N\colon \mathbb{R}[X] \rightarrow \mathbb{R}$, such that : $N$ is a norm of $\mathbb{R}$-vector space and $N(PQ)=N(P)N(Q)$ for all $P,Q \in \mathbb{R}[X]$.
...
1
vote
2answers
111 views
Hilbert dual space (inequality and reflexivity)
Let $V \subset H$ where $H$ is Hilbert space. Let $T:H^* \to V^*$ be the canonical map that restricts the domain of a functional in $H$ so that it's a functional in $V$.
How do I show that
$$\lVert ...
6
votes
2answers
477 views
Does every $\mathbb{R},\mathbb{C}$ vector space have a norm?
Is there a canonical way to define on any vector space over $\mathbb{K}=\mathbb{R},\mathbb{C}$ a norm ?
(Or, if there isn't, can someone give me an example of a vector space over $\mathbb{K}$ that is ...
5
votes
2answers
68 views
Is there a lower-bound version of the triangle inequality for more than two terms?
The triangle inequality $|x+y|\leq|x|+|y|$ can be generalized by induction to $$|x_1+\ldots+ x_n|\leq|x_1|+\ldots+|x_n|.$$
Can we generalize the version $|x+y|\geq||x|-|y||$ to $n$ terms too? I need ...
4
votes
1answer
228 views
Equivalence of reflexive and weakly compact
In a normed space $X$ is there an equivalence between these two proposition?
1) $X$ is reflexive;
2) $B$, the unit ball of $X$, is weakly compact.
4
votes
3answers
201 views
Simple application of Stone-Weierstrass
I was looking for a simple application of the Stone-Weierstrass theorem.
First I thought that if $X$ is any compact measure space then Stone-Weierstrass implies that $C_c(X)$ is dense in $L^p$.
But ...
3
votes
1answer
369 views
A question on linear transformation
Let $T$ be a linear transformation on $\mathbb{R}^{4}$ whose standard
matrix is
$$\left(\begin{array}{rrrr}
1 & -1 & -1 & -1\\
1 & 1 & 1 & -1\\
1 & 1 & -1 & 1\\
...
2
votes
1answer
53 views
Banach spaces and their unit sphere
Let $X$ be a normed vector space.
Show that if a subsequence of a Cauchy sequence converges, then the whole sequence converges.
Use the part 1 to show that $S = \{x\in X : \|x\| = 1\}$ is complete ...
2
votes
4answers
187 views
Sum of closed convex set and unit ball in normed space
Let $(X, \|.\|)$be a real normed space. Let $A$ be a closed convex suset of $X$ and $\mathbb{B}$ a unit ball in X, i.e.
$$
\mathbb{B}=\{x\in X: \|x\|\leq 1\}.
$$
I would like to ask whether ...
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 ...
9
votes
2answers
174 views
If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?
Let $V,W$ be two normed vector spaces (over a field $K$). Then their product $V \times W$ with the norm $\|(x,y)\| = \|x\|_V + \|y\|_W$ is a normed space.
Using this norm it's easy to show that if ...
4
votes
1answer
255 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 ...
4
votes
2answers
215 views
If you know the convergent sequences, how do you know the open sets?
I have a homework problem which I feel should be simple but is actually surprisingly tricky. This is why I love math sometimes....
Let $X$ be a normed linear space. Suppose $\|\cdot\|_1$ and ...
4
votes
1answer
189 views
Characterization of normed vector spaces of finite dimension
I have this problem:
Let $E$ be a normed vector space. $S=\{x\in E : ||x||=1\}$. Show that if $S$ is compact then $\dim E$ is finite.
This follows directly from the Riesz's lemma, but in the notes ...
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 ...
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
1answer
64 views
Proper Linear Subspaces
Let $X$ be a normed space. I want to prove that for any proper linear subspace $M$, there exists a point $x$ with $||x||=1$ such that $inf\{||x-y||:y\in M\}>1-\epsilon$ for arbitrary epsilon. ...
2
votes
1answer
75 views
inequality between norms of vector
Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$.
It is known that $\|a\|_{\infty}\leq \|a\|_2\leq \sqrt n\|a\|_{\infty}$.
Let $k<<n$. For which kind of vectors the following would be true:
$$
...
2
votes
0answers
165 views
Question about proof of Stone-Weierstrass
I would like to know if I understand the details in the proof of Stone-Weierstrass (in $\mathbb R$) so I'd like to post it here in my own words. Can you please check it and tell me if it's correct? ...
2
votes
1answer
56 views
Norms on inner product space over $\mathbb{R}$
Definition of the problem
Let $\left(E,\left\langle \cdot,\cdot\right\rangle \right)$
be an inner product space over $\mathbb{R}$. Prove that for all $x,y\in E$
we have
$$
\left(\left\Vert ...
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 $y = \theta$.
...
1
vote
2answers
61 views
Inequality regarding weak-* convergence
Let $X$ be a normed linear space, $\psi \in X^{*}$ and $\displaystyle \{\psi_n\}_{n \in \Bbb N}$ a sequence in $X^{*}$. Show that if $\displaystyle \{\psi_n\}_{n \in \Bbb N}$ converges weak-${*}$ to ...
1
vote
3answers
314 views
Show that the unit sphere is strictly convex
I can prove with the triangle inequality that the unit sphere in $R^n$ is convex, but how to show that it is strictly convex?
1
vote
1answer
83 views
Sum of weak star closed set and compact weak star set
Let $X$ be a real Banach space and $X^*$ be its dual space.
Let $C$ be a weak$^*$ closed subset in $X^*$ and $D$ a compact weak$^*$ in $X^*$.
I would like to ask whether $C+D$ is closed weak$^*$ in ...
1
vote
1answer
57 views
Euclidean sections of normed spaces with known cotype
I'm having trouble digesting the proof of Theorem 9.6 in Milman and Schechtman's classic book "Asymptotic theory of finite dimensional normed spaces" (pg. 54). I'm new to functional analysis, so this ...
1
vote
1answer
118 views
$C_0(X)$ is a closed subspace of $C_b(X)$
Can you tell me if my proof is correct? Thank you!
Claim: $C_0(X)$ is a closed subspace of $C_b(X)$
Proof: We have to show that $C_0(X)$ contains all of its limit points. Let $f(x)$ be a limit point ...
1
vote
1answer
90 views
Typo in lecture notes?
The following is an example in my lecture notes:
"Let $X$ be a locally compact topological space (that is, a topological space in which every point has a
compact neighborhood). Then
$C_0(X)=\{f \in ...
1
vote
0answers
480 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 ...
1
vote
1answer
382 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 ...
