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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If $E\subset X^{*}$ is bounded, then so is its weak* closure

If $X$ is a Banach space and $E\subset X^{*}$ is norm-bounded, I've shown that its weak* closure is also norm-bounded using Alaoglu's theorem. But perhaps using Alaoglu's theorem is not necessary? ...
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1answer
28 views

Trying to show that $(c_0, \| \cdot \|_s)$ is strictly convex, where $\| x \|_s = \underset{i = 1}{\overset{\infty}{\sum}} \frac{1}{2^i} | x_i |$

I'm trying to show that $ (c_0, \| \cdot \|_s) $ is a strictly convex space, where $$ \| x \|_s = \underset{i = 1}{\overset{\infty}{\sum}} \frac{1}{2^i} | x_i |,$$ $ x = (x_1, x_2, ..., x_i, ...) \in ...
3
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1answer
60 views

Application of Banach Steinhaus theorem

Let $X,Y$ be Banach spaces and let $A:X\rightarrow Y,B:Y^*\rightarrow X^*$ be linear maps. Show that if for all $x\in X,y\in Y^*,y^*(Ax)=By^*(x)$, then both $A$ and $B$ are continuous
1
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1answer
80 views

Uniform limit in definition of second order directional derivatives

If $f:E\rightarrow F$ is twice differentiable at $x\in E$, do we then have $$\lim_{h,k\rightarrow 0}\frac{A_x(h,k)-f''(x)(h)(k)}{\|h\|\|k\|}=0$$ where $A_x(h,k):=f(x+h+k)-f(x+h)-f(x+k)+f(x)$? This is ...
4
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1answer
71 views

Fundamental proof of Taylor's theorem using little-o notations

Is there a fundamental proof of Taylor's theorem using little-o notation? I assume $f:E\rightarrow F$ as a mapping between Banach spaces and write $(h^i)$ for $(h,\ldots,h)$ ($i$ times iterated). ...
0
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1answer
32 views

GNS Construction on non-unital algebra

STATEMENT: If A has a multiplicative identity 1, then it is immediate that the equivalence class $ξ$ in the GNS Hilbert space H containing 1 is a cyclic vector for the above representation. If $A$ is ...
2
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0answers
37 views

Making values of functionals simultaneously positive [duplicate]

We work with real Banach spaces. Let $X$ be an infinite-dimensional Banach space and let $(f_n)$ be a sequence of norm-one functionals on $X$ that are linearly independent. Can we find an element ...
0
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0answers
24 views

Inductive/Projective Limits of Topological Algebras

It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance, For $k \ge 0$ and $K_n$ compact ...
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0answers
29 views

Supports of elements in $\ell_\infty$

It is idle curiosity that makes me ask so apologies for no motivation. Let $X$ be an infinite-dimensional, closed subspace of $\ell_\infty$. Can we find a non-zero element $x\in X$ such that ...
1
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1answer
60 views

Reflexive Banach spaces and the Cantor Intersection property

Let $(E, \lvert \cdot \rvert )$ be a reflexive Banach space. As a consequence of the Banach-Alaoglu Theorem and Mazur's Lemma we have that every nested non-increasing sequence $ C_1 \supset C_2 ...
0
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1answer
33 views

Definition of stable under isomorphism

I am reading a paper 'A coding of separable Banach spaces' and it says 'we identify a family of separable Banach spaces which is stable under isomorphism with a subset of $C=\{$closed subspaces of$ ...
2
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0answers
68 views

Prove $\left(\operatorname{Lip}\left([0,1]\right),\lvert\lvert\cdot\rvert\rvert\right)$ is a Banach Space

We denote by $\operatorname{Lip}\left([0,1]\right)$ the collection of all Lipschitz functions on $[0,1]$. We know that a function $f:[0,1] \to \mathbb R$ is called Lipschitz if there exists $K>0$ ...
2
votes
3answers
71 views

Why if $T$ is not continuous, then for each $n\in N$, there exists $x_n\in X$ such that $||Tx_n||\ge n||x_n||$

If $X$ $Y$ are Banach space, $T$ is a linear operator between them. I don't understand the following statement: If $T$ is not continuous, then for each $n\in N$, there exists $x_n\in X$ such that ...
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0answers
24 views

an inequality in Banach algebra [duplicate]

Let $(V, \| \ \|)$ be a Banach algebra. Given two elements $x,y\in V$ satisfying $xy=yx$, prove that $$ ...
0
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1answer
27 views

Invertibility of operators related to Markov processes in Ethier-Kurtz

Lemma 2.3 of the book by Ethier and Kurtz (first edition, I believe) defines $$ g_n := (\lambda - A)(\lambda_n - A)^{-1}g $$ for some fixed $ g $ but I see no guarantee that $(\lambda_n - A)^{-1} g ...
1
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1answer
40 views

What's the difference between a Banach Algebra and a CStar Algebra?

I'm currently looking at going into a PhD program in mathematics and need to decide on a specialization. In meeting with my advisor he pushed me into looking at Cstar algebra's based on my interests. ...
0
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1answer
61 views

orthonormal basis for $L^2(\mathbb{R})$.

So if we consider $L^2(\mathbb{R})$ as an Hilbert space with inner product $(\cdot ,\cdot)$. Define $\psi_n(x)=e^{-\frac{x^2}{2}}H_n(x)$ where $H_n(x)$ is the Hermite Polynomials. Then how do you show ...
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0answers
28 views

Question on calculus of functions that take values in Banach Spaces

I've recently started a course on PDEs, and one of the first constructions we made was the following. Say we have a PDE: $$F(t,x_1,\dots,x_n,u,u_t,u_{x_1},\dots,u_{x_n},\dots) = 0$$ with $u(t,x): ...
2
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1answer
52 views

Injective tensor product

We know that $c_0\check\otimes c_0=c_0(c_0)$ where $\check\otimes$ is the injective tensor product. is the following still true? $$c_0\check\otimes l^\infty = l^\infty(c_0).$$ Thank you for your help ...
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1answer
68 views

Stone-Weierstrass: Summary

This is just a summary. Theorem Given a compact domain $\Omega$. Regard the function space: ...
2
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1answer
39 views

A consequence of the open mapping theorem

We let $f$ be a bounded and surjective linear map from the Banach space $X$ onto the Banach space $Y$ and put $$ r_0=\inf\{r: f(B^X(0,r)\supset B^Y(0,1)\}. $$ Using the open mapping theorem, I have ...
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1answer
31 views

Stone-Weierstrass: Lattice

This is just a prework. Given a compact domain. Regard the function space: $$\mathcal{C}(\Omega,\mathbb{R}):=\{f:\Omega\to\mathbb{R}:f\text{ continuous}\}$$ Clearly it is an algebra: ...
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0answers
35 views

question about density of Sobolev spaces

I have a short question about density of spaces. Consider: $C_c^{\infty}(0,1)=\{f\in C^{\infty}(0,1); supp(f)\subset (0,1)\;\text{compact}\}, $ ...
4
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2answers
121 views

Arzela-Ascoli: Proof?

Problem Given a compact domain. Regard the function space: $$\mathcal{C}(\Omega):=\{f:\Omega\to\mathbb{C}:f\text{ continuous}\}$$ Consider a bounded family: ...
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0answers
18 views

Completeness of $C^1(E,F)$

Can someone just confirm the following, I think I already managed to prove it but it seems so important that I want to know if I did some mistake... If $f_i:U\subseteq E\rightarrow F$ is a sequence ...
1
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1answer
26 views

A question about complement of a closed subspace of a Banach space

Let $X$ be a Banach space and $M$ be a closed subspace of $X$. Suppose that there exists a subspace $N$ of $X$ such that $X=M\oplus N$. Does it imply that $N$ is closed ? I know that not every ...
0
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1answer
38 views

Stone-Weierstrass: Literature

Short question... Does someone know some textbook, a paper or notes that treats: An algebra of functions with identity that separates is dense within a function space.
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1answer
46 views

Convexity of a certain function connected to the norm

Suppose that we are given two vectors $x,y$ in a normed space $X$. Can we prove in general that the function $$t\mapsto \|x-ty\|$$ is convex? It is certainly the case if the normed space has ...
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1answer
29 views

Complex Measures: Function Space

Given a locally compact Hausdorff space $\Omega$ and a Banach space $E$. Denote functions with compact support by: $$\mathcal{C}_0(\Omega,E):=\{F\in\mathcal{C}(\Omega,E):\operatorname{supp}F\subseteq ...
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1answer
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Is this linear map bounded?

This question asks to prove the following: Let $X$ and $Y$ be Banach spaces. If $T: X \to Y$ is a linear map such that $f \circ T \in X^*$ for every $f \in Y^*$, then $T$ is bounded. The ...
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0answers
27 views

Interchanging limits in proof of a norm

I'm trying to prove that $\|f\|=\sup \{|f(x)|x\in [a,b]\}+ \sup \{|f'(x)|:x\in [a,b\}\}$ defines a norm on $C^1[a,b]$ (the vector spaces of continuously functions on [a,b] with continuous first ...
0
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1answer
12 views

Absolute value inequality for Pettis integral

Let $f:[a,b]\rightarrow E$ be absolutely continuous and Pettis integrable, i.e. there exists $I_f\in E$ such that $x^*(I_f)=\int x^*\circ f$ for $x^*\in E^*$. Because $f$ is absolutely continuous, ...
2
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0answers
34 views

Bochner integral vs regulated integral

I'm reading Serge Lang's Real And Functional Analysis and at some point he introduces the regulated integral in order to prove the Fundamental Theorem Of Calculus (in the context of Banach Spaces), or ...
2
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1answer
73 views

infinite-dimensional Banach spaces has linear subspaces of finite-codimension that are not closed

I want to show that In every infinite-dimensional Banach spaces there are linear subspaces of finite-codimension that are not closed . There is a hint for this question that says use Zorn ...
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1answer
54 views

Resolvent: Norm

Given a Banach space. Consider a closed operator: $$T:\mathcal{D}(T)\to E:\quad T=\overline{T}$$ Due to the Neumann series it holds: $$R(\lambda):=(\lambda- ...
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1answer
26 views

Proving a certain subset is closed in $L^1$

In an exam, I was asked to prove that, if $A=\{f\in L^1([0,1]):\int_0^1|f(x)|^2\mathrm{d}x\leq1\}$, then $A$ is closed in $L^1$. I tried this approach. $A$ is closed iff for all $f_n\to f$ in $L^1$ ...
3
votes
2answers
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Maximal Ideal Spaces

STATEMENT: Let $C_b(\mathbb{R})$ be the $C^*$-algebra of bounded continuous functions on $\mathbb{R}$. Let $A$ be the $C^*$-subalgebra of $C_b(\mathbb{R})$ generated by $C_∞(\mathbb{R})$ together with ...
0
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1answer
22 views

Bochner Integral: Lebesgue Point

This thread is just a note. Given an euclidean space and a Banach space. Consider Bochner integrable functions: $$F\in\mathcal{B}(\mathbb{R}^d,E):\quad\int\|F\|\mathrm{d}\lambda<\infty$$ Then ...
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1answer
28 views

Banach algebra of homomorphisms

Let $E,F$ be Banach spaces. Is it always true that $\mathrm{Hom}(E,F)$ is Banach algebra ?
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1answer
29 views

If $x_j \to x $ and $||Tx_j ||\le k $, show that $||Tx|| \le k $ for a continuous linear operator $T$

Let $T $ be a continuous linear operator. Suppose ${x _j } $ is a sequence in some Banach space $X $, with limit $x $, such that $||Tx _j || \le k $. Show that $||Tx ||\le k $ Well I suppose that I ...
0
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1answer
42 views

A $\| \cdot \|_2$-closed subspace of $C[0,1]$ is always Banach.

I was doing a problem and I realize that if I prove that given a Y $\|\cdot \|_2$-closed vectorial subspace of $\mathcal{C}[0,1]$ is Banach, I'm done. Well, what I've been trying is establishing a ...
2
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1answer
26 views

Are ideals generated by separable subspaces separable?

Suppose that $X$ is a compact Hausdorff space and take a sequence $(f_n)$ in $C(X)$ such that the ideal generated by $(f_n)$ is proper. Must this ideal be separable as a Banach space? It looks to me ...
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1answer
90 views

Compact Operators: Weak Convergence [duplicate]

Problem Given Banach spaces $X$ and $Y$. Consider a compact operator $C\in\mathcal{C}(X,Y)$. Then weak convergence is turned into strong convergence: $$x_n\rightharpoonup x\implies Cx_n\to Cx$$ I'd ...
0
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1answer
32 views

Is $C[0,1]$ complemented in $B[0,1]$?

A closed subspace $V$ of a normed space $X$ is called complemented if there exist a closed subspace $E$ such that $E \oplus V=X$, equivalently if there exist a continuous operator $T:X\rightarrow V$ ...
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1answer
48 views

Every finite-dimensional subspace is one-complemented

Let $X$ be a Banach space. It is known that if every closed subspace of $X$ is one-complemented, then $X$ is isometrically isomorphic to a Hilbert space. Now if every finite-dimensional subspace of ...
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1answer
40 views

Compact Approximation

This is meant as lemma for: Approximation Property Given a Banach space $E$. Denote compact operators by $\mathcal{C}(E)$. Consider a compact domain $C\subseteq E$. Then there is a compact ...
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1answer
25 views

Specific question on Banach space over nonarchimedean field

Let K be an nonarchimedean field. We let $Ban(K)$ denote the category of $K$-Banach space with continuous linear maps and let $C$ be the category of normed K-Banach spaces ($V, ||$ $ || $) ...
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If a map between separable Banach spaces has closed graph, does it have a point of continuity?

It is well known that the closed graph theorem does not directly extend to nonlinear maps: even for functions from $\mathbb{R}$ to $\mathbb{R}$, having closed graph does not imply continuity. But ...
2
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1answer
44 views

Is my proof that $\ell^p$ is Banach correct?

I would like to show that $\ell^p$, for $p \in [1, \infty)$ is Banach, using the following fact: Given a normed linear space $X$, $X$ is Banach if and only if absolute convergence in $X$ implies ...
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0answers
27 views

Spatial tensor product of operator spaces

If $X$ and $Y$ are Banach spaces and $\otimes_\varepsilon$ denotes the injective tensor product, then in general $\otimes_\varepsilon$ does not respect quotients unless we map into ...