Tagged Questions
0
votes
1answer
33 views
How to expand equation inside the L2-norm?
I want expand an L2-norm with some matrix operation inside.
Assume I have a regression $Y=X\beta+\epsilon$.
I want to solve (meaning expand),
$$\displaystyle\|Y-X\beta \|_{2}^2$$
Should I do:
1)
...
2
votes
1answer
58 views
Inequality for norms
Let g(x, y) be function on $X\times Y$. Show that for all $p\geq q$
$$
\|\,\|g\|_{L^q(Y)}\,\|_{L^p(X)}\leq \|\,\|g\|_{L^p(X)}\,\|_{L^q(Y)}
$$
Thsnk you.
3
votes
3answers
55 views
Example of two norms on same space, non-equivalent, with one dominating the other
I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...
1
vote
2answers
39 views
Linear functional $\mathscr{L}(E,F)$
Let $\mathscr{L}(E,F)$ denote the space of all linear functionals from $E \to F$.
Let $\mathscr{C}(E,F)$ denote the space of continuous linear functionals from $E \to F$. My question:
How to prove ...
1
vote
1answer
43 views
Study the equivalence of these norms
I have two Hilbert spaces $H_1$ and $H_2$ and I consider a set of functions $f$ which decompose as $f=g+h$ with $g\in H_1$ and $h\in H_2$. I know that this decomposition is unique.
So I define the ...
2
votes
0answers
22 views
Are these two definitions for dual norm equivalent
Suppose there is a p norm, also is a function $\left\|x\right\|_p$, the dual function at $z$ of it is,
$$\sup\limits_x\left\{ z^{T} x-\left\|x\right\|_p\right\}$$
The second is,
$$
\sup\limits_x ...
2
votes
0answers
29 views
Finding an orthornormal basis given a bilinear form
Take $ \langle \cdot,\cdot\rangle$ to be a nondegenerate symmetric bilinear form (which may not be positive definite). Take $\Vert x\Vert = \vert \langle x,x\rangle \vert^{1/2}$. I want to show that ...
3
votes
0answers
50 views
What are norms used for?
These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
0
votes
1answer
32 views
Norm of a function, Smoothness Penalization
I am seeking for some intuition why norm (for any reasonable norm on functions) of a function is smaller if the function is smoother.
2
votes
1answer
53 views
finding operator norm $T_N$
How do find the operator norm of , $T_N\colon c_0\to \Bbb R$ given by
$T_N(y):=\sum_{j=1}^Nx_jy_j$ ,when $N$ is a integer .
TIA
1
vote
1answer
44 views
Does $\|z\|=(\sqrt{|x|}+\sqrt{|y|})^{\frac{1}{2}}$ define a norm on $\mathbb R^2$
Does $\| z\|=(\sqrt{|x|}+\sqrt{|y|})^{\frac{1}{2}}$, with $z=(x,y)\in\mathbb R^2$, define a norm on $\mathbb R^2$?
2
votes
1answer
51 views
A specific linear operator between Banach spaces
Let B be the Banach space $B=(C[0,1],\|\cdot\|_{\infty}$) and let $\{\xi_i\}\in l^\infty$.
Let $T:l^1\rightarrow B$ be the linear operator given by: $(Ta)(x) = \sum_n\xi_na_nx^n$.
I have three ...
3
votes
2answers
93 views
Two proofs concerning Hölder's inequality
I am studying functional analysis and I have come across two statements which can be proven by using Hölder's inequality, but I don't know how/why. Reminder: let $a\in l^p$ and $b\in l^q$, then:
$$
...
6
votes
2answers
114 views
equivalence of norms
I would like a little help here:
I have two defined norms over $C^{1}([0,1])$ :
$\| A(f)\|=|f(0)|+\max_{x\in[0,1]}{|f'(x)|}$
$\| B(f)\|=\int_0^1|f(x)|dx+\max_{x\in[0,1]}{|f'(x)|}$
I already ...
0
votes
3answers
160 views
How to find the norm of this bounded linear functional?
Let $C^\prime[a,b]$ denote the normed space of all continuously differentiable real (or complex) valued functions defined on the closed, bounded interval $[a,b]$ in $\mathbf{R}$ with the norm defined ...
0
votes
0answers
24 views
Which of the following functions are norms? [duplicate]
For $x=(x_1,x_2)$, which of the following functions on $\mathbb{R}^2$ are norms?
a.) $A_1(x) = 7\mid x_1\mid + 3\mid x_2\mid$,
b.) $A_2(x) = \text{max}\lbrace\mid x_1\mid^2,\mid x_2\mid^2\rbrace$,
...
0
votes
1answer
23 views
A limit superior question in the context of the Neumann series
I'm trying to understand a step in the proof that the Neumann series converges:
Let $X$ be a Banach space and $T\in L(X)$ (the space of bounded, i.e. continuous linear maps $X\to X$). It is known ...
0
votes
0answers
74 views
Bound for the infinity norm of a vector
Let $x$ be a vector in $R^n$ such that its support $\operatorname{supp}(x)=\frac{n}{2}$. Show that $\|x\|_{\infty}\leq \sqrt{\frac{2}{n}}$.
Thank you.
0
votes
1answer
44 views
Norm of normal Operator A
I just found the following equality $||A||=sup_{\lambda\in\sigma(A)} |\lambda|$
My question: What exactly means $\sigma(A)$ and why this is true ?
I always thouht the only way to get the ...
2
votes
1answer
63 views
Cauchy-Schwarz for metrics with arbitrary signatures
When the norm of a vector is always greater than or equal to zero, the Cauchy-Schwarz inequality holds, but what if we look at a metric with an arbitrary signature? Then the inner product of a vector ...
2
votes
2answers
95 views
equivalence of norms and direct sum
Let $(X,\|\cdot\|_X) $ be an infinite dimensional Banach space. Suppose that you can write $X=V\oplus W$. Write $x=v+w$ and define $(V\oplus W,\|x\|_\oplus :=\|v\|_X+\|w\|_X)$. Show that ...
4
votes
2answers
118 views
What is the reason norm properties are defined the way they are?
Suppose we have a complex vector space $V$.
A norm is a function $f : V \rightarrow \mathbb{R}$ which satisfies
(i) $f(x) \ge 0$ for all $x \in V$ (Positivity - Non-Negativity)
(ii) $f(x + y) \le ...
3
votes
0answers
75 views
Is $\rho(f) := \mu(|f| \geq \epsilon), \epsilon >0$ some kind of generalized norm?
Let $\rho(f) := \mu(|f| \geq \epsilon)$ where $\epsilon >0$ is a constant and $\mu$ is a measure and $f$ is a measurable function. Then
$f\equiv 0 \rightarrow \rho(f) = 0$
when $|a| \neq 0$, ...
1
vote
1answer
64 views
Small question regarding norms and Holder conjugates.
I'm trying show that if $p,q$
are Holder Conjugates then: $$\forall\, a\in\mathbb{R}^{n}:\,\Vert a\Vert_{q}=\max_{x\in\mathbb{R}^{n},\,\Vert x\Vert_{p}=1}\left<a,x\right>$$
Where ...
5
votes
1answer
108 views
Norm in a dual space
If $f \in X^*$, with $X^*$ the dual space consisting of all linear bounded functionals on a linear normed space $X$. With the norm defined as $||f||_{X^{*}} = \sup_{||x|| \leqslant 1} |f(x)|$. Why ...
2
votes
1answer
86 views
Quotient norm on $X\backslash M$
I have $X=(C([0,1]),||.||_1)$ where $||f||_1=\int_{0}^{1}|f(t)|dt$ and $M=\{f\in C([a,b]): f(0)=0\}$. Now I have three questions:
1) Is the quotient norm a norm on the quotient space X\M ?
What I ...
2
votes
1answer
86 views
Equivalence of two norms
Define two norms as following: $$
\left\Vert f\right\Vert _{1}={ \max_{0\leq x\leq1}\left|f\left(x\right)\right|}
, \quad\text{ and }\quad \left\Vert f\right\Vert ...
2
votes
1answer
85 views
Equivalence between $Lip \ norm$ and $C_1 \ norm$.
Let $f\in C^1([a,b])$. Prove that $\|f\|_{C^1} = \|f\|_{Lip}$.
By definition of Lip norm and $C^1$ norm, it is equivalent to prove that $\|f'\|_{\infty}=Lip(f,(a,b))$, where the second member is the ...
2
votes
1answer
306 views
proving $L^\infty$ norm inequality (disprove $\Vert f\Vert_\infty\le\sqrt{n}$)
There are three parts in this question, I've done the first two but not sure about the third one. Also see $L^2$ norm inequality.
In the third part, I am asked to show that if $W$ is a ...
1
vote
1answer
71 views
The openness of the set of positive definite square matrices
Let $\mathbb{R}^{n\times n}$ be the vector space of square matrices with real entries.
For each $A\in \mathbb{R}^{n\times n}$ we consider the norms given by:
$$
\displaystyle\|A\|_1=\max_{1\leq j\leq ...
13
votes
1answer
385 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 $P,Q \in \mathbb{R}[X]$.
...
3
votes
1answer
396 views
Relations between p norms
The p-norm is given by $||x||_{p} = (\sum_{n=1}^{\infty
}|x_{n}|^{p})^{1/p}$. For $0 < p < q$, it can be shown that $||x||_{p} \geq ||x||_{q}$ (1, 2). It sppears that in $R^{n}$ a number of ...
1
vote
2answers
121 views
My “wrong” comparison between $\ell^2$ and $\ell^1$
For sure, $\ell^2$ is larger than $\ell^1$， because for $|x|<1$, $|x|^2<|x|,$ that is, $||x||_2\leq||x||_1.$
But using Cauchy-Schwartz inequality, I get a "wrong" comparison:
...
1
vote
5answers
114 views
Is closure of convex subset of $X$ is again a convex subset of $X$?
Today I was going through my text book exercise and got hold of the following question.
Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
2
votes
1answer
123 views
$C^1 [0,1]$ with different norm
If the space $C^1 [0,1]$ is equiped with norm $\Vert \cdot\Vert_1$,where
$$
\Vert f\Vert_1=|f(0)|+\Vert f'\Vert _{C}=|f(0)|+\sup_{t\in [0,1]}|f'(t)|
$$ for any $f\in C^1 [0,1]$, is this space Banach?
...
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:
$$
...
1
vote
2answers
96 views
Does $(f,Tg)_{L^2}$ define an inner product space?
To my understanding inner product
$$(f,g)_{L^2(\mathcal{D})} = \int_\mathcal{D} f(\boldsymbol{x})g(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x},~~\mathcal{D} \subset \mathbb{R}^N$$
defines an inner ...
3
votes
1answer
90 views
The trace norm cannot be increased by composing with a unitary operator
$\newcommand{\tr}{\operatorname{tr}}$
I was reading a proof for the statement $|\tr(US)|\leq |\tr(S)|$, for every endomorphism $S$ on a complex vector space $H$ and every unitary operator $U$ on the ...
2
votes
1answer
311 views
Proof that every normed vector space is a topological vector space
The topology induced by the norm of a normed vector space is such that the space is a topological vector space.
Can you tell me if my proof is correct? Of course we have to show that addition and ...
0
votes
1answer
95 views
How to find/parameterize vector perpendicular to circle of constant $\ell_p$ norm
This should be very easy, but I can't get my head around it: given $1\leq p < \infty$, and a point x with $\|x\|_p = 1$, how do I get a (or the unit, or any) vector which is perpendicular to the ...
2
votes
1answer
98 views
Limit inferior taken on the norm of a sequence
Let $E$ a normed vector space and let $(x_n)$ be a sequence in $E$. Suppose that $x_n$ converges weakly (i.e. wrt the weak topology) to $x$.
Why is it that from the inequality
$$
|f(x_n)| \leq \|f\| ...
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 ...
2
votes
1answer
396 views
Multiplication operator norm
I'm having some (hopefully small) issues computing the norm of an operator. Firstly, the problem,
For $f\in L^\infty[0,1]$, define $M_f: L^2[0,1]\to L^2[0,1]$ by $M_f(g)(x) = f(x)g(x)$. Show that ...
3
votes
1answer
368 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\\
...
1
vote
1answer
272 views
Cauchy sequence in a normed space
Let $V$ be a real vector space. Suppose that $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ are two norms on $V$ which are equivalent.
I suspect the following to be true.
Let $(x_n)_{n=0}^\infty$ ...
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 ...
