3
votes
2answers
75 views

Showing a set is nowhere dense in $C([0,1])$

Let $E_n$ be the set of all $f \in C\big([0,1]\big)$ for which there exists $x_0 \in [0,1]$ (depending on $f$) such that \begin{align*} \lvert\, f(x)-f(x_0)\rvert \leq n\lvert x-x_0\rvert, ...
3
votes
0answers
27 views

Every Cauchy net is convergent [duplicate]

Prove that in a Banach space every Cauchy net is convergent. I have trouble to prove this, please help.Thanks Edit:Let $A$ be a directed set and $\{f_{\alpha}\}_{\alpha\in A}$ is a net in $X$ ...
-1
votes
3answers
96 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 ...
0
votes
0answers
50 views

Proof that a set is open.

Let $(\Lambda_i)_{i\in I}$ a collection of linear operators from $X$ (Banach space) to $Y$ (Normed space). Let $\alpha : X \rightarrow [0,\infty]$ be the function $\alpha(x):=\sup_{i \in I} ...
2
votes
1answer
48 views

Prove that this space is not Banach

Let $\Omega\subset\mathbb{R}^n$ be an open, bounded set with boundary $\partial\Omega$ of class $C^1$. $$\mathcal{A}:=\{u\in C^2(\bar\Omega):u=0\text{ on }\partial\Omega \}$$ endowed with the scalar ...
1
vote
1answer
30 views

Functional defined on the space of functions with compact support

Let $X=C_c(\mathbb{R})$, the space of functions with compact support, normed with the max norm. Define $\Gamma: X \rightarrow \mathbb{R}$ as: $$ \Gamma f= \int_{- \infty}^{\infty} f(t)dt ...
0
votes
1answer
34 views

Exercise on isometry

Let $X$ be a Banach space and $T$ a linear bounded operator defined on $L(X,Y)$ with $Y$ a normed space. If $T$ is an isometry then $TX$ is a closed subspace of $Y$. I considered a sequence $y_n$ ...
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 ...
0
votes
0answers
21 views

Polar set and adjoint operator

I'm trying to understand the following statement: A bounded operator between Banach spaces $u:X\rightarrow Y$ satisfies: $$\textrm{inf}\{\;||u(B_X \cap S)||:S\subset X,\,\textrm{codim}S=k \; ...
0
votes
1answer
24 views

Contraction map

I have a general question about the properties of contractive/non-contractive maps. Assuming I have a map $F$ which I know is not a contraction. Can it map a ball back to itself, i.e. for some ...
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
28 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 ...
2
votes
1answer
47 views

Equivalence of norms given continuous identity

It is known that $\parallel \; \parallel_{1}$ & $\parallel \; \parallel_{2}$ are equivalent norms over $X$ if there are $A,B>0$ such that $A\parallel x \parallel_{1} \leq \parallel x ...
0
votes
1answer
43 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
57 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 ...
4
votes
1answer
74 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 ...
3
votes
0answers
47 views

The dual of the Annihilator

Let $X$ be a Banach space, and $I$ be a closed subspace. Then it's known that $(X/I)^*=I^{\perp}$. My question is what is the second dual of $X/I$? or what is the dual of $I^{\perp}$ ? If we know ...
1
vote
1answer
28 views

Continuity of an application between function spaces.

I'm trying to prove the following statement... Let $f:[a,b] \times \mathbb{R} \to \mathbb{R}$ a bounded and continuous function, $t_{0} \in [a,b]$, $x_{0} \in \mathbb{R}$, $r>0$ and $$B= \{ x ...
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 $\{f_{n}\}$ ...
5
votes
1answer
77 views

Closure of compact sets in Banach space

Let $(X,\vert\vert\cdot\vert\vert)$ be a Banach space. For each $k\in\mathbb{N}$ let $A_k\subseteq X$ be compact and $r_k\in\mathbb{R},r_k>0$, such that $$A_{k+1}\subseteq \{x+u\vert x\in A_k ...
0
votes
1answer
25 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
38 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 ...
0
votes
0answers
25 views

Is the set, $\{f\in Y: \left\|f\right\|_{Y} \leq C \ \text{and} \ \left\|(|f|)\right\|_{X}\leq C \}$, closed in $(Y, ||\cdot||_{Y})$?

Put, $X= \text{The space of "nice" complex valued functions on } \mathbb R; \text{that is}, f:\mathbb R \to \mathbb C;$ so that $X$ is Banach space with respect to the norm ...
0
votes
1answer
36 views

when one can expect $\left \| |x|^{2}x- |y|^{2}y \right \| \leq \frac{1}{2} \left \| (x-y) \right \|$?

Suppose $(X, \left\|\cdot \right \|)$ is Banach algebra with the property that $\left\| (|x|^{2}x) \right\| \leq \left\|x\right \|^{3};$ for every $x\in X.$ Let $y_{0}\neq 0 \in X$ and fix it; and ...
1
vote
1answer
38 views

Proof check for completeness

I'd like to see if the proof I have is adequate. Statement. Let $X$ and $Y$ be Banach space, the product $X\times Y$ is a vector space under coordinate operations with norm $$ \|(x,y)\| = \|x\|_X ...
5
votes
1answer
154 views

Homeomorphisms between infinite-dimensional Banach spaces and their spheres

As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere. Unfortunately I do not have his book but I want to know is this theorem true without ...
3
votes
1answer
90 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 ...
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: ...
1
vote
1answer
36 views

differences between Banach spaces and $\Bbb R^n$.

Can you please tell me, what are the biggest differences between Banach spaces and $\Bbb R ^n$? I am trying to understand the Frechet derivative.
3
votes
1answer
69 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 $Y$ ...
0
votes
1answer
46 views

Check in which points is f differentiable.

Check in which points is f differentiable: $f: l_{ \Bbb R}^{ \infty} \ni x \rightarrow ||x||_{ \infty} \in \Bbb R$ I started with this: $ \lim_{||h||_{\infty} \to 0} ...
0
votes
1answer
49 views

Show that the following functional is Frechet differentiable in Hilbert space

I need to show that the following functional is Frechet differentiable: $$ f(u) = \|u\|^2_{H} \ \ \text{in a real Hilbert space} \ \ H $$ Solution: As far as I understand, I need to take a Taylor ...
1
vote
1answer
82 views

Show that the following functional is Frechet differentiable

I am new to this and I need to show that the following functional is Frechet differentiable: \begin{equation} f(u) = \sin(u(1)) \ \ in \ \ C[0,1] \end{equation} What I have already done: ...
3
votes
2answers
70 views

Does $L^1$ contain a subspace isomorphic to $c_0$?

Can any $L^1$ space, say $L^1(\mathbb{R})$, have some subspace isomorphic to $c_0$? I guess not but I don't see an argument right now.
1
vote
0answers
58 views

$L^{\infty}$ is p-concave

I want to show that the $L^{\infty}$ on a Banach lattice $X$ is $p$-concave with $M_{(p)}(L^{\infty})=1$. Where $L^\infty=L^\infty(X,\mathcal{M},\mu)$. Recall that a Banach lattice $X$ is said ...
1
vote
1answer
57 views

Absolutely convergent sums in Banach spaces

Let's say a sum of elements in a Banach space is absolutely convergent if even the sum of the norms converges, i.e. $\sum_{i=1}^\infty ||x_i|| \le \infty$. This condition implies that the sum of the ...
0
votes
1answer
182 views

Quotient norm and actual norm

I have a question about the proof that $X\backslash U$ is a Banachspace if $X$ is one and $U$ is closed. In my book it is said, that for $x_k \in X$ and a series $\sum_{k=1}^{\infty}||[x_k]||< ...
1
vote
0answers
128 views

Application of fixed point theorem for an integral equation.

Consider the following non-linear integral equation: $f(x)=\lambda \int_a^b K(x,y;f(y))dy + \phi(x)$ with $K$ and $\phi$ continuous functions , such that $K$ satisfies the Lipschitz condition on the ...
0
votes
1answer
61 views

Proof of the nonexistence of an identity $\phi$ involving convolution

The Banach space $L^1(\mathbb{R}^n)$ is an algebra with a product (convolution) which is both commutative and associative. But this algebra does not have a multiplicative identity. An attempt to show ...
0
votes
1answer
119 views

Let $E$ be a Banach space, prove that the sum of two closed subspaces is closed if one is finite dimensional

Let $E$ be a Banach space and let $S$ and $T$ be closed subspaces, with dim$\space T<\infty$. Prove that $S+T$ is closed. To prove that $S+T$ is closed I have to show that for any limit point $x$ ...
0
votes
1answer
48 views

Limit of a sequence that is Cauchy with respect to a family of seminorms

Definition: Let $p$ be a seminorm on some space $X$. A sequence $(x_n)$ in $X$ is said to be Cauchy with respect to $p$ if for any $\epsilon > 0$, there is an $N > 0$ such that $p(x_m - x_n) ...
1
vote
3answers
224 views

From analysis of realvalued functions to analysis of Hilbert/Banach-valued functions

Does anybody know of a text (doesn't matter which form: article, book etc. - anything's welcome) in which it is described which result from real analysis also hold for Hilbert/Banach spaces ? I'm ...
1
vote
1answer
244 views

Derivative Bilinear map

I wanted to calculate the derivative of a continuous bilinear map $B: X_1 \times X_2 \rightarrow Y$. (Does anyhere know whether there is a generalisation of the notation $L(X,Y)$ that you use for the ...
1
vote
1answer
128 views

Reflexivity of a Banach space without the James map

The reflexivity of a Banach space is usually defined as having to be enforced by a particular isometric isomorphism. Namely the map that takes each element to the evaluation, which is already an ...
0
votes
0answers
100 views

Questions about the Gateaux derivative

We defined that a function is Gateaux differentiable, if all directional derivatives exist. I just wanted to check, whether I got a few things right: Now I wanted to ask, whether it is true that if ...
4
votes
1answer
1k views

Prove that $C^1([a,b])$ with the $C^1$- norm is a Banach Space

Consider the space of continuously differentiable functions, $$C^1([a,b]) = \{f:[a,b]\rightarrow \mathbb{R}|f,f' \text{are continuous}\}$$ with the $C^1$-norm $$||f|| = \sup_{a\leq x\leq ...
2
votes
2answers
732 views

Understanding the Frechet derivative

What is the relationship between the normal 'high school' concept of a derivative and a Frechet derivative? According to wikipedia the Frechet "extends the idea of the derivative from real-valued ...