0
votes
0answers
30 views

The restriction of an open bounded linear operator

I need some help with this question. Let $X$ be a Banach space and $T:X \to X$ be a bounded linear operator. Suppose that $T$ is open, and $X_0$ be a closed subspace of $X$. The restriction $T_0$ of ...
-1
votes
1answer
27 views

Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Also note that though I were able to resolve the first problem the second one is still ...
-1
votes
3answers
90 views

Differentiation in Banach spaces

Let $E$ be a Banach space, and $F:=L(E,E)$, with $L(E,E)$ the set of continuous linear funtions in $E$. Prove that the function $\exp: F → F$, defined by ...
3
votes
1answer
86 views

Prove that $L^1(\mathbb{N})$ is a Banach space.

I'm trying to prove that $L^1(\mathbb{N}) := \left\{ (x_n)_{n=1}^{\infty} : \sum\limits_{n=1}^{\infty}\left|x_n\right| < \infty \right\} $, the space of all sequences over the field $\mathbb{C}$ ...
2
votes
2answers
27 views

Bounded, surjective linear operator between Banach spaces

How can I show that for a given surjective linear operator $T: X \to Y$ between Banach spaces, if there exists an $\epsilon > 0$ such that $||Tx|| \geq \epsilon||x||$ for all $x \in X$, then $T$ is ...
1
vote
1answer
25 views

$g(T)$ bounded implies $T$ bounded, if $T$ is linear and $g$ is bounded linear functional

Suppose $X$ is a Banach space and $y$ is a normed linear space and $T :X\to Y$ is linear map such that for every bounded linear functional $g\in Y^*$ we have $g( T)$ is bounded . Show that $T$ is ...
2
votes
1answer
21 views

Baire Category for monotonic sequence

Let X be a non-empty complete metric space and let $\{f_n:X\to \Bbb R\}^\infty_{n=1}$ be a sequence of continuous functions with the following property: for each $c\in X$, there exists an integer ...
3
votes
2answers
44 views

Computing the norm of operator when space is equipped with sup norm and $L^1$ norm

Let $\phi $ be the linear functional $\phi (f)=f(0)-\int^1_{-1}f(t)\:\mathrm{d}t$ a.Compute the norm of $\phi$ as a functional on Banach space $C[-1,1]$ with sup norm. b.Compute the of $\phi$ as a ...
0
votes
2answers
42 views

$||\phi||=1$ and $|\phi (x)|=||x||$

a.Let $E$ be a non-zero Banach space and show that for every $x\in E$ there is $\phi \in E^*$ such that $||\phi ||=1$ and $|\phi (x)|=||x||$ b. Let E and F be Banach spaces,let $\pi: E\to F $ be a ...
0
votes
1answer
14 views

I need help showing something is a linear continuous operator.

Define $T:C([0,1])\rightarrow C([0,1])$ by $T(f)(x)=f(0)+\int_0^xtf(t)dt$ I want to show that $T$ is a continuous linear operator.Showing the linear part is easy enough, but I am not quite sure how to ...
-2
votes
1answer
108 views

Continuity of $|.|$ in $W^{1,p}_0$

please i dont understand this proof We suppose that $u\rightarrow u$ on $W^{1,p}_0$ i dont understand why we must use the weak compactness and the uniform convexity of $W^{1,p}_0$ ? Thank you
0
votes
0answers
28 views

Let $V:= (C([0,1]),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the function $f_n$ and show that V is not a Banach Space.

The Assignment: Let $V:= (C([0,1]),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the continuous function $f$ for all $n \in \mathbb{N}$: $$f_n: [0,1] \rightarrow \mathbb{R}, \ ...
2
votes
1answer
82 views

Have they solved this exercise correct?(banach space, function space).

Please look at this exercise: It is the last question I have a problem with Here is the solution: They say that $\|I(1)\|=\sup|g(t)|$. But isn't $\|I(1)\|=\int_0^1g(t)dt$? If so, what is the ...
0
votes
2answers
12 views

Sequence of continuous functions with unitary norm

I want to construct a sequence $f_n$ of continuous functions in $[0,1]$ such that $||f_n||=1$ (so a bounded sequence) and $||f_n-f_m||=1$ (it doesn't have any convergent subsequences). The norm is ...
1
vote
1answer
46 views

Nontrivial functionals on $l^\infty$ vanishing on $c_0$

I understood that the dual of $c_0$ is a proper subspace of the dual of $l^\infty$, by Hahn-Banach theorem. But how can I find functionals in $(l^\infty)^*$ vanishing on $c_0$?
2
votes
1answer
38 views

Banach space under the Lip norm

Let $(X,d)$ be a compact metric space. A function $f:x\to \Bbb R$ is said to be Lipschitz continuous if $$\|f\|_d = \sup\left\{\frac {|f(x)-f(y)|}{d(x,y)}:x,y\in X,x\neq y\right\}< \infty.$$ Denote ...
2
votes
1answer
42 views

lower semi continuous on Banach space implies locally bounded?

Let $(X, \|\cdot\|)$ be a Banach space; and $f:X\to [0, \infty)$ is lower- semi continuous on $X.$ My Question is: Can we expect $f$ is bounded in some open subset of $X$ ? [If answer is ...
1
vote
1answer
27 views

question involving $\ell^p$ spaces

I would like to prove the following: I am not asking for a solution! I would simply like a bit of guidance, I can't seem to get started on this problem. Would the proof involve using weak ...
0
votes
0answers
30 views

A Banach space $X$ is relfexive iff $X^*$ is reflexive. [duplicate]

I am trying to prove the following: A Banach space $X$ is reflexive iff $X^*$ is reflexive. Thus far, I have proven the forward direction: Let $J_X:X\mapsto X^{**}$ be the mapping defined by ...
1
vote
1answer
36 views

Weak convergence of subsequence in Hilbert spaces

Prove that if $x_n$ is a sequence in $H$ (Hilbert space) with $\sup_n||x_n||\le1$, then there is a subsequence $\{x_{n_j}\}$ and an element $x$ of $H$ with $||x||\le 1$ such that $x_{n_j}$ converges ...
0
votes
1answer
40 views

Frechet derivative of the sup norm function on $C[0,1]$

I know that the derivative in the title does not exist for any x but I do not have a clue why. Could someone explain why this derivative DNE at 0 and I can figure out why it does not exist at x? ...
1
vote
0answers
28 views

Biorthogonal functionals continuous? [duplicate]

If I have a Schauder basis $(x_n)$ of a Banach space $X$. Such that for every $x = \sum_{i=1}^{\infty} a_i x_i$ for a unique sequence $(a_i) \subset \mathbb{R}$. Is it obvious that the functionals ...
6
votes
3answers
56 views

Counterexamples for Hölder's inequality when $p$ and $q$ are not conjugate.

Hölder's inequality shows that, when $$ \frac{1}{p} + \frac{1}{q} = 1,$$ and $f\in L^p$ and $g\in L^q$, then $$\Vert f\,g\Vert_1 \le \Vert f \Vert_p \Vert g \Vert_q.$$ Is there an example of this ...
0
votes
0answers
26 views

Does Hilbert space with countable dimensions exist? [duplicate]

If there is a Hilbert space with infinite dimensions, can it have countably infinite dimensions? And does Banach space with countable dimensions exist?
3
votes
1answer
173 views

Proof of equicontinuous and pointwise bounded implies compact

I tried to prove the Arzela-Ascoli theorem: Let $X$ be a compact Hausdorff space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb R$ endowed with the sup norm $\|\cdot ...
4
votes
1answer
71 views

Vector spaces isomorphic, then dual spaces isomorphic

If we know that there is a (topological) isomorphism between two Banach spaces $X,Y$ called $\phi \in L(X,Y)$. Then the appropriate isomorphism between the dual spaces $X',Y'$ is given by $\phi' \in ...
2
votes
1answer
36 views

$f_{n}$ converges in $L^{1}$ and $\|g_{n}\|_{L^{1}(\mathbb R)}\leq \|f_{n}\|_{L^{1}(\mathbb R)}\implies {g_{n}}$ converges in $L^{1}(\mathbb R)$?

Let $f_{n}, g_{n}\in L^{1}(\mathbb R)$ (Lebesgue space). Suppose there exist $f\in L^{1}(\mathbb R)$ such that $\|f_{n}-f\|_{L^{1}(\mathbb R)}\to 0$ as $n\to \infty;$ that is, the sequence $\{f_{n}\}$ ...
0
votes
0answers
46 views

Completeness is not preserved under homeomorphism

I am trying to give a counterexample which will make it clear that homeomorphism does not preserve completeness. I have an easy one in mind ($(0,1)$ and $\mathbb{R}$) but I have just thought that ...
1
vote
0answers
24 views

The Haar basis ,proof of orthonoramality.

please i have this problem and i known how to prove completeness but do not know how to prove that it is orthonormal. I will appreciate it if anyone can help me. Given that $n\geq1$ write ...
0
votes
1answer
23 views

understanding a proof involving equivalence of norms in finite dim. linear normed spaces

I am reading the proof of the theorem shown below (from Linear Functional Analysis by Rynne and Youngson). I can't figure out why the part I highlighted in red is true. I understand why $S$ is compact ...
0
votes
1answer
54 views

Banach space problem

I came across the following problem: For an open set $U$ in $\mathbb{R^n}$ we define the set of all k-times continuously differentiable functions $f:U\rightarrow \mathbb{R}$ for which $D^\alpha f$ ...
1
vote
1answer
34 views

Bounded operators on unit ball and equicontinuity

Let $X$ be a Banach space and $B$ be the closed unit ball contained in $X$. Let $\{T_{\alpha}\}$ be a family of bounded linear operators from $B$ to $V$, a normed vector space. My question is: Suppose ...
1
vote
1answer
50 views

Are $C^0[a,b]$ and $C^0[0,1]$ isometrically isomorphic?

Consider $C^0[a,b]$ and $C^0[0,1]$, each equipped with the $L^1$-Norm. Are these (out of curiosity) isometrically isomorphic?
0
votes
1answer
66 views

This is Banach space?

Let me denote by $C_{0}(\mathbb{R})$ the set of continuous functions which tend to zero at + and - $\infty$. I am wondering if it is true that $(C_{0}(\mathbb{R}), \Vert . \Vert_{\infty})$ is a ...
0
votes
2answers
116 views

Density of $C^{1}_{0}(\mathbb R)$ in $L^{\infty}(\mathbb R)$

I am looking for a counterexample to $C^{1}_{0}(\mathbb R)$ ( $C^1$ functions with compact support) is dense in $L^{\infty}(\mathbb R)$? Is there some easy counterexample showing that this latter is ...
3
votes
1answer
89 views

Why the image of a linear map is not always a Banach space?

I have a question: Let's think about the map $T:V \rightarrow \text{ran}(T)$ and $V$ be a Banach space. Then we have that this is the same as the quotient map $[T]:V \rightarrow V/\ker(T)$ where the ...
0
votes
1answer
44 views

In which Banach spaces of functions do the convergence almost everywhere and boundedness of norms imply the weak convergence?

See the title. If I am not mistaken, this is true for $L^2$-spaces but is false for $L^1$-spaces. Say, should the space be reflexive?
1
vote
1answer
71 views

Continued matrices-valued function

Given $d<k$. Let ${\cal M}_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}_{d\times k}(\mathbb{R})$ is a continuous ...
1
vote
0answers
154 views

Is it possible to isometrically embed a bounded metric space into a Hilbert space?

My practical intention is described in this question, I am already doing this in my experiments with some success but I want a mathematical formulation in place not only to justify but also to verify ...
0
votes
0answers
17 views

Given data, approximations in a metric space for moving into a normed vector space isometrically.

Please see this question and this answer. Here $f_x(y)$ is approximated by $$x_v = [d(x,K_1),d(x,K_2),....d(x,K_N)]$$ by choosing to consider distances from $x$ to only certain points $K_i$ and ...
1
vote
2answers
115 views

Collecting things that are preserved by (isometric) isomorphisms between normed spaces

I would like to collect a list of things that are preserved under isomorphisms and isometric isomorphisms. The reason is that I hope to get a better perception of their importance in Functional ...
2
votes
1answer
39 views

Proof of an equivalence in Hilbert spaces

Let $H$ be a Hilbert space. Prove that the following are equivalent: a) the algebraic dimension of $H$ is finite; b) each closed, not empty subset $C$ has an element of minimum norm (that is the ...
1
vote
1answer
53 views

norm of Frechet derivative in point.

Let $ E = \mathcal B([0,1], \Bbb R) $ with supremum norm. Now I can define function $ F:E \ni f \rightarrow ||f||^2 - f(0) \in \Bbb R$ My task: 1)Show the differentiability of $F$ in: $ f_0: ...
3
votes
1answer
66 views

Hyperplane which can separate two closed convex set

Define: $$E=:\ell^1(R)=\{x=(x_n)_n: \|x\|_1=\sum_{n=1}^{\infty}|x_n|<\infty\}$$ We have two closed convex sets $X,Y$ as subsets of normed vector space $\ell^1(R)$ with $X\cap Y=\emptyset$ and $Y$ ...
2
votes
0answers
136 views

Second derivative of a composite function

Say, we have three Banach spaces $X, Y, Z$ and $g:X \to Y, \ \ f:Y \to Z$ are twice (Fréchet) differenciable. The question is: what is $(f \circ g)''$? Since $(f \circ g)'':X \to ...
7
votes
4answers
360 views

$\ell^p$ is not isometric to $\ell^q$

The problem is this: if $1\le p<q<\infty$ then $\ell^p$ and $\ell^q$ are not isometric (as Banach spaces). This is an exercise but I'd like to see an elegant proof.
5
votes
0answers
105 views

When does Gâteaux imply Fréchet? [duplicate]

Speaking of the relation between Gâteaux and Fréchet, authors usually point out that $$\text{Fréchet} \implies \text{Gâteaux}$$ and then give a counterexample to illustrate that the converse doesn't ...
1
vote
1answer
40 views

Pulling Operator Inside Integral

Say $Y$ is a Banach space and you have a family of continuous/bounded operators $L_{x}: Y \rightarrow Y$ for $x\in \mathbb{R}$ and say you have an bounded, smooth map $f(x):\mathbb{R}\rightarrow Y$. ...
3
votes
1answer
51 views

Convergence in $L^1$ of a sequence of functions

I have to see if the following sequence of functions is convergent in the space $L^1[(0,\infty)]$ $$f_n(x)= n\frac{\exp\left(-\frac{n}{2x^2}\right)}{x^3}$$ By definition, $f_n(x)$ is convergent in ...
2
votes
1answer
58 views

What is the dual of $(\mathbb{R}^n,∥\cdot∥_{\infty})$

I know that the dual of $(\mathbb{R}^n,∥\cdot∥_{p})$ is $(\mathbb{R}^n,∥\cdot∥_{q})$ with $\frac{1}{p}+\frac{1}{q}=1$ But does this also hold when $p=\infty$ and if so what is the proof?