A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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strict subadditivity of the norm in uniformly convex Banach Spaces.

So here is my question, I would like to prove the following, Let $(X,||\cdot||)$ be a uniformly convex Banach Space. Then the norm is strictly subadditive i.e $\forall x,y\in ...
2
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1answer
61 views

the principle of uniform boundedness and $l^p$ space

If $1<p<\infty$ and $\{x_n\}\subset l^p$, then $\sum_{j=1}^\infty x_n(j)y(j)\to 0$ for every $y\in l^q$, $\frac{1}{p}+\frac{1}{q}=1$, iff $\sup_n||x_n||_P<\infty$ and $x_n(j)\to 0$ for every ...
2
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1answer
59 views

Dual of injective tensor norm is not projective tensor norm

Let $A$, $B$ are two Banach space, on the algebraic tensor space $A$ $\odot$ $B$, we can define the projection(maximal) tensor norm $\gamma$ and injective(minimal) tensor norm $\lambda$. For the ...
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1answer
36 views

existence of an “inverse” adjoint in Banach Spaces

Let X,Y be Banach spaces and S a bounded operator $S: Y' \rightarrow X'$ where $'$ denotes the dual space or the adjoint operator depending on what it is on. Then $$ \exists \ \ T \in B(X,Y) : T'=S ...
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1answer
50 views

1<p<q, norm inequalities

Prove or disprove the following. (Recall that $c_{00}$ is the space of sequences with only finitely many nonzero entries.) Conjecture. Let $1<p<q<\infty$. Then there exists a function ...
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1answer
66 views

Is it true that $0<0$ is a part of the proof for a unique fixed point of a contraction map?

I think it isn't, but my friend told me that this statement is true, he said that $0<0$ is even part of the proof for a unique fixed point of a contraction map. I google it but I couldn't find ...
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2answers
23 views

Bounding the distance between $L_\infty$ and $L_2$ for a continuous function

Consider a set of continuous (or even differentiable) functions $f_i(x)$, all defined for $x\in [a,b]$ for $i=1\ldots,N$. Can one define a uniform constant $c$ (which may depend on $f$) such that ...
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1answer
75 views

Does existence of a non-continuous linear functional depend on Axiom of Choice?

Well, it is easy to construct a non-continuous linear functional on an arbitrary infinite-dimensional vector space (assuming Choice, and taking a basis etc.). I think it is intuitive to say that: ...
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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 ...
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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 ...
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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 ...
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1answer
40 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 ...
2
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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 ...
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1answer
27 views

Approximating a mean on $L^{\infty}(G)$ by a norm $1$ non-negative $f\in L^{1}(G)$

Let $G$ be a locally compact group. A mean $M$ on $L^{\infty}(G)$ is a continuous linear functional on $L^{\infty}(G)$ such that $1 = \langle 1 , M\rangle = \|M\|$. My Exercise: Show that the set ...
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1answer
33 views

Does completing a normed space commute with taking quotients?

Let $X$ be a normed vector space and $Y \subset X$ a closed subspace. We consider the quotient $X / Y$ and equip it with the quotient norm. Then we may form the completion $\overline{X / Y}$. We ...
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1answer
81 views

Banach subspace of $C(X)$ [duplicate]

Let $X$ be compact and suppose that $Y$ is a Banach subspace of $C(X)$. If E is a closed subset of $X$ such that for every $g\in C(E)$ there is an $f\in Y$ with $f_{|_{E}}=g$. Show that there is a ...
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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? ...
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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 ...
2
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1answer
77 views

Two questions on Banach-valued spaces of integrable functions

Let $V$ be a Banach space with dual $V'$ and suppose that $V$ is included into the Hilbert space $H$ so that the inclusion is continuous and dense. Then after identification $H = H'%$ we have $V ...
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0answers
48 views

the dual space of $L^p$ [duplicate]

I am reading some preliminary material to develop a good background in order to study PDE and I came across the following fact The dual space of $L^p$ is $L^q$ where $q$ is the Holder's Conjugate of ...
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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 ...
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1answer
36 views

Is the Neumann series a compact operator?

Let $X$ be an infinite dimensional Banach space and $A:X\to X$ be a compact operator with the operator norm $\|A\|<1$. Then $I-A$ is invertible and the Neumann series $$ S_N = \sum_{k=0}^N A^k $$ ...
2
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1answer
51 views

Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ...
5
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0answers
151 views

Subsequences of a basic sequence

Suppose ($x_n$) is a basic sequence in a Banach space $X$, and $Y$ is a closed, infinite co-dimensional subspace of the closed span of $(x_n)$. Can we always find a subsequence ($y_n$) of ($x_n$) such ...
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0answers
27 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?
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1answer
180 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 ...
2
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1answer
54 views

Banach spaces: Convergence in terms of the Schauder basis.

Let $X$ be a Banach space. Suppose $X$ has a normalized Schauder basis $\{x_n\}_{n \in \Bbb N}$. Let $\{y_n\}_{n \in \Bbb N}$ be a sequence in $X$ converging to $0_X$. For each $n \in \Bbb N$, let ...
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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 ...
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1answer
75 views

Norm of a projection in $L_p$

Let $\mu$ be a probability measure on $[0,1]$ and let us define the projection $P$ on $L^p(\mu)$: $$P \; \colon f \mapsto \mathbb{E}(f){\bf 1}$$ (where ${\bf 1}$ is the constant 1 function). What is ...
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1answer
29 views

Do stochastic processes form a Banach space?

I'm interested in solving a particular integral equation: $$g(X) = \int_0^1 K(X,p)f(p) \ dp$$ where $f(p)\in L^1([0,1])$ and $X$ is a stochastic process of finite length; i.e. a collection of random ...
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0answers
36 views

Examples of semigroups of contractive Fourier multipliers but not positive?

Can you show me a concrete an example of semigroup $(T_t)_{t\geq 0}$ of Fourier multipliers such that each operator $T_t$ induces a contractive Fourier multiplier $T_t\colon L^p(\mathbb{T}) \to ...
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1answer
62 views

Why is $I+T$ invertible for this rank-one operator $T$?

I am working with the following lemma given in the book Topics in Banach Spaces Theory: Let ${(x_n)}_{n=1}^{\infty}$ be a basic sequence in Banach Space $X$. Suppose that there exists a linear ...
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0answers
41 views

uniform equivalence to unit vector basis of $\ell_p$

Let $(e_n)$ be the unit vector basis of $\ell_p$, $1\leq p<\infty$. It is well-known that if $(x_n)\subset\ell_p$ is seminormalized and weakly null then it contains a subsequence equivalent to ...
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1answer
28 views

How to show that $L^\infty(0,T;L^\infty(\Omega))$ is complete?

How to show that $L^\infty(0,T;L^\infty(\Omega))$ is complete? I did the usual: let $u_n$ be a Cauchy sequence, then we get $$\text{esssup}_t \;\text{esssup}_x |u_n(x,t)-u_m(x,t)| \leq \epsilon$$ Now ...
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1answer
36 views

Unit ball weakly* compact

So here is my problem, I am trying to understand the proof of, $X$ Banach space $\Rightarrow$ the unit ball in $X^*$ is weakly* compact. The proof uses Tychonoffs Theorem to conclude the ...
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0answers
36 views

If the quotient of a subspace of a banach space is finite, is it a closed subspace?

Given a Banach space B,V is a subspace of B,if B/V is finite dimension,then is it enough to show that B is closed? Thanks!
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1answer
32 views

Is Limit and Norm interchangable in Banach Spaces

Suppose $X$ is a Banach space, and $\{x_n \}\subset X$. Does it then hold that $\lim \|y-x_n\|=\|y-\lim x_n \| $ ?
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1answer
54 views

The image of an bounded and linear operator between two banach spaces is closed if the image is finite codimension?

So here is my problem, I would like to prove the following, Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded operator. Then $TX$ is closed if it is of finite codimension i.e ...
6
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1answer
108 views

Quotient map, quotient topology in Banach spaces

In Lindenstrauss and Tzafriri's Classical Banach Spaces I an operator $T:X\to Y$ is called a quotient map if the $\overline{TB_X}=B_Y$ where $B_X$ and $B_Y$ are the unit balls in Banach spaces $X$ and ...
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0answers
105 views

How to get a grip on codimensions

I am trying to find a proof for the following problem: Let $X,Y$ be Banach spaces $A,B:X \rightarrow Y$ are bounded linear operators $Ran(A)$ is closed, and $\dim(\mathrm{Ker}(A))$ or ...
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0answers
25 views

Confusion with proving that some subspace of a Banach-Space is closed

So here is my problem, I am trying to show that, Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded map. Then $T(X)$ is closed if $Y/T(X)$ is of finite dimension. While ...
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0answers
112 views

Strongly convex and Frechet differentiable function in reflexive Banach space

We first recall two definitions about strong convexity and Frechet differentiability in normed space. Let $(X, \|.\|)$ be a normed space and $f:X\rightarrow\mathbb{R}$ be a function. (a) $f$ is said ...
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2answers
202 views

Integration in Banach spaces - interesting, nice and non-trivial examples needed

I am interested in $\textbf{Integration in Banach spaces}$. Here is a little motivation for my question: Let $\left(X,\|\cdot\| \right)$ be a Banach space, $a,b \in \mathbb{R}$ with $a<b$ and $f ...
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1answer
52 views

Spectrum and characters: could anyone please check my proof

I tried to prove the following: Let $A$ be a commutative non-unital complex Banach algebra and $\chi : A \to \mathbb C$ a character. Then $$ \sigma (a) = \{\chi (a) : \chi \in \Omega (A) \} ...
2
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1answer
76 views

Prove $Tf$ is continuous, $T$ is a contraction and find a solution to the integral $f(x)$

On the Banach space $(C([0,1]), ||.||_\infty)$, consider the operator given by $$(Tf)(x)=\int_0^1 x^2tf(t) dt+1.$$ a.) Prove that $Tf$ is continuous for all $f\in C([0,1])$. b.) Show that $T$ is a ...
2
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1answer
45 views

$(\ell_p,\|.\|_{\infty})$ Banach or separable

Is $(\ell_p,\|.\|_{\infty})$ for $1\leq p<\infty$ a Banach or separable space? is there any fast way of proving it without checking separability or completeness with the usual way?
3
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0answers
106 views

On the weak convergence in reflexive Banach space

Consider the following proposition: Proposition 1. Let $X$ be a reflexive Banach space and suppose that $\{x_n\}$ is a sequence in $X$ that is bounded and has at most one weakly sequentially cluster ...
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1answer
82 views

Which are the conditions for a Lorentz space $L^{p,q}$ to be o-c?

Which are the conditions for a Lorentz space $L^{p,q}$ to be ord. continuous? ( A Banach function space is o-c $\equiv$ Increasing sequences of order-bounded positive functions converge in norm). ...
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1answer
52 views

The equivalence of norm continuous and weak continuous

Let $X$ be a Banach space and $\phi: X \rightarrow C$ be a bounded linear functional. Then $\phi$ is weakly continuous is equivalent to $\phi$ is norm continuous, right? Why?
3
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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 ...