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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Gateaux and Frechet derivatives on $\mathbb{R}^2$.

I have the following problem: Let $f:\mathbb{R}^2 \to \mathbb{R}$ be defined by: \begin{equation} f(x,y)= \frac{x^3y}{x^4+y^2}, \quad x \neq 0, y\neq 0 \end{equation} and: \begin{equation} ...
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Prove that $(\mathbb R^n,\|\cdot\|_3)^*= (\mathbb R^n,\|\cdot\|_{1.5})$ with full details [on hold]

How can one prove that $(\mathbb{R}^{n},\|\cdot\|_3)^*= (\mathbb{R}^{n},\|\cdot\|_{1.5})$ with full details. How to go about using Hölder's inequality? I know that these are complete inner ...
3
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Hamel basis and Banach spaces

Suppose $X$ is a linear space and $X$ has a Hamel basis with uncountable number of elements. Does there exist a norm on $X$ such that $X$ is a Banach space with respect to this norm?
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18 views

Bijective bounded linear operator is invertible

The following is an exercise from Halmos book "A Hilbert space problem book" : Exercise: If $H$ and $K$ are Hilbert spaces, and if $A$ is a bounded linear transformation that maps $H$ one to one and ...
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Definition of Szlenk index, $w*$ closed set.

I am reading a paper and it has the following: Let $X$ be a separable Banach space. Given $\epsilon>0$, and a $w^*$- closed subset $P$ of $B_{X^*}$, we let $P_\epsilon'=\{x^*\in P \mid $ for all ...
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1answer
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Do any $\ell^{p}(\omega)$ have the extension property?

Definition 1. A metric space (normed space) $X$ has the extension property exactly in case, for all finite $A\subseteq X$ and isometric (linear isometric) $f:A\rightarrow X$, there exists an extension ...
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Verification of a fact, separable Banach spaces, closed subset.

I am reading a proof and it says to verify the following: Suppose $Z$ is a separable Banach space and $F$ is a closed subset of $Z$. Let $\mathcal{O}$ be a countable basis of open subsets of $Z$. We ...
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strong epis in the category of banach spaces with linear contractions

In Borceux's Handbook volume 1, page 145, the strong epis in the category of Banach spaces with bounded linear maps of norm <= 1 is characterized as the maps whose restriction on the unit balls is ...
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1answer
39 views

Lp spaces - example of function

Can you give me an example of function which: $$f \in L^{p}[a,b]$$ but $$f \not\in L^{\infty}[a,b]$$ $L^{\infty}[a,b]$ is space of essentially bounded function at interval $[a,b]$ $1 \le p < ...
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1answer
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Prove that $S:F \to \mathcal{L}(E,F)$ is a topological isomorphism

Let $E$ and $F$ be normed spaces, $E \neq \{ 0 \}$ and $x_0 \in E \backslash \{ 0 \}$, $x_0 \in E'$ such that $x_0'(x_0)=1$. Prove that the function $S: F \to \mathcal{L}(E,F)$, $S(y)=T_y$ defined ...
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Compact metric implies uncountable w*-dense set

I am reading a proof of the following: Let $X$ be a separable Banach space. The Szlenk index is countable iff $X*$ is separable. In the proof of => it uses the following: If $X^*$ is not separable, ...
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2answers
30 views

question about the proof that the set C[a,b] with uniform norm is complete

I am trying to understand the proof that the set of continuous function is complete under uniform/supremum norm. First, suppose we have a Cauchy sequence of continuous functions ${f_n(t)}$ with ...
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18 views

Essential Ideals and Denseness

Let $A$ be a $C^\ast$ algebra. I am wondering if essential ideals in $A$ are dense. It seems like they should, but I don't know how to show it.
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A question in Banach space

Given $A\in\Bbb R^{m\times n}$, define $$\gamma_2(A)=\min_{XY=A}\max_{i,j}\|x_i\|_{\ell_2}\|y_j\|_{\ell_2}$$ where rows of $X$,columns of $Y$ given by $\{x_i\}_{i=1}^m,\{y_j\}_{j=1}^n$ respectively. ...
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Converse of uniform boundedness principle

The uniform boundedness principle says if we have a collection of bounded linear operators $\Gamma$ from a banach space $X$ into a normed vector space $Y$, which is pointwise bounded on $X$, i.e. ...
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1answer
26 views

Weak convergence + compactness = strong convergence? [duplicate]

Let $X$ be a Banach space and $K$ a compact subset of $X$. If $(x_n)_n$ is a sequence such that $x_n\in K$ for all $n$ and $(x_n)_n$ converges weakly to some $x\in X$, i.e. $x^*(x_n)\to x^*(x)$ for ...
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1answer
25 views

Bijection between a Hilbert and a Banach space

I am working on a question where I have to show that for a Hilbert space $\mathscr{H}$, and closed linear subspace $Y \subset \mathscr{H}$, $\mathscr{H}/Y$ is a Hilbert space, isomorphic to ...
2
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1answer
63 views

The Banach–Mazur distance for finite-dimensional $\ell_p$

Let $\ell_p$ denote the usual infinite-dimensional sequence space, and if $n\in\mathbb{Z}^+$ then we let $\ell_p^n$ denote its $n$-dimensional counterpart. Conjecture. Let $1\leq p<\infty$. ...
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WOT convergence in the unit ball of B(X)

My questions is (probably) related to: On separable Hilbert space $H$, weak operator topology is metrizable on bounded parts of $B(H)$ Does the theorem quoted in the above question, together with ...
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1answer
28 views

Question on banach space over an extension of $\Bbb{Q}_p$

Let $G$ be a compact locally $\Bbb{Q}_p$ analytic group. Let $E$ be a finite extension of $\Bbb{Q}_p$ with ring of integers $O$. Let $M$ be a $O[G]$ module. I was reading an article which says : ...
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Are there hypersurfaces with connected complement in a Banach space?

In $\mathbb{R}^n$ it is well-known that a smooth hypersurface $M$ (closed as a subset of $\mathbb{R}^n$) is the zero locus of a global smooth function (whose gradient is nonzero on $M$); from this one ...
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1answer
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preserving problem

Define: $$\begin{align*} \varphi &:L^2[0,1] \to L^2[0,1], \\ &(\varphi f)(x)=‎\int_0^x f(t)dt. \end{align*}‎$$ Is it true that, if ‎‎$‎‎B$ is a dense ‎subset ‎of ‎‎$‎‎L^2[0,1]$, then ...
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1answer
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On the cardinality of $\mathbb R \times …\aleph_1 {times}$ and $\mathbb R \times …2^{\aleph_0} \space {times}$

I think I can prove that closure of every countable set in any metric space has cardinality at most $\mathcal c=2^{\aleph _0}$ . So if a metric space is separable i.e. has a countable dense subset $A$ ...
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1answer
32 views

Asymptotically nonexpansive mapping that is not nonexpansive

Let $C$ be a nonempty subset of a Banach space $X$. A mapping $T:C\to C$ is said to be asymptotically nonexpansive mappings if for all $n\in \mathbf{N},$ there exists a positive constant $k_n\geq1$ ...
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1answer
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Completion of a vector space inside a given Banach space

Let $X$ be a normed vector space and $Y$ be a Banach space such that $X$ is continuously embedded into $Y$ (this will be denoted by $X\hookrightarrow Y$ in the sequel). Is it always possible to find ...
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1answer
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asymptotically nonexpansive mappings

Let $C$ be a nonempty subset of a Banach space $X$. A mapping $T:C\to C$ is said to be asymptotically nonexpansive if for all $n\in \mathbf{N},$ there exists a positive constant $k_n\geq1$ such that ...
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Is the following metric space is a complete metric space?

We have $X=\ell^1$, which contains sequences, which are absolutely convergent, and $d(a_n,b_n) = \sum_{k=1}^{\infty}|a_k-b_k|$. Is this metric space complete or not?
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1answer
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Scales of Banach Spaces: Literature?

Does someone know a nice reference for: $$E^{-s'}\hookrightarrow E^{-s}\hookrightarrow E^0=E\hookrightarrow E^s\hookrightarrow E^{s'}\quad(s\leq s')$$ (I need a more abstract view; less focus on PDE.) ...
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Reference request: linear operators into $L^\infty$ can be extended presrving the norm. [duplicate]

Suppose $X$ is a normed linear space and $Y\subset X$ a linear subspace. I remember that any linear map $L\colon Y\to L^\infty(\Omega)$ can be extended to a linear map $\tilde{L}\colon X\to ...
2
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1answer
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inequality in a banach space

Let $(X,\left\| \cdot \right\|)$ be a Banach space. For each $i=1,\cdots,n$, let $a_i\in X$ and $\alpha_i\in\mathbb{R}$. Suppose that $0\leq\alpha_i\leq M$ for all $i=1,\cdots,n$. Question: Is it ...
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Does a Banach space always contain an element of arbitrarily large norm?

Let $X$ be a Banach space. Or say, even just a normed linear space. Let $N \in \mathbb N$. Does there exist $x \in X $ with $\|x\| \ge N$? If $X$ is a Banach space then its unit ball is also a ...
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1answer
45 views

Quotient Rings of Algebras

So let us take the commutative Banach algebra $B=\mathscr{l}^1(\mathbb{Z}_n)$ over $\mathbb{R}$ with convolution as multiplication $(f\ast g)(x)=\underset{y\in \mathbb{Z}_n}{\sum} f(y)g(x-y)$. I know ...
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Do three valued basis vector elements lead to the fastest discrete Fourier transforms?

When sin() and cos() are approximated to 1, 0 and -1 in the basis vectors in a real or discrete Fourier transform the basis vectors have a lot of elements of zero or in common leading to an algorithm ...
4
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Comparison between weak convergences in Banach spaces

Let $X$ be a Banach space and let $Y=BC(\mathbb{R},X)$ be the Banach space of all bounded continuous functions from $\mathbb{R}$ to $X$ equipped with the supremum norm. Let $(f_n)_n$ be a sequence of ...
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Inverse Function Theorem. On the classical method of proof.

The proof most commonly of the Inverse Function Theorem seen in textbooks of relies on the Banach fixed-point theorem. QUESTION. It is possible to say, using historical references, which ...
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1answer
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Show that $X$ is Banach space and describe $X^*$.

Let $X=L^2(\mu)\times L^2(\mu)=\{(f,g)|f,g\in L^2(\mu)\}$ be the linear space normed by $\|(f,g)\|=(\|f\|_2^3+\|g\|_2^3)^{1/3}$. Show that $X$ is Banach space and describe $X^*$. My Work: We ...
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About a technique used in the proof of Hahn-Banach Theorem

Recall Hahn-Banach (cf. Kreyszig's book) : If $X$ is a real vector space with a sublinear functional $p$ and if $f$ is linear on a subspace $Z$ with $p(z)\geq f(z),\ z\in Z$, then there exists an ...
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Convergence in Bochner space (don't follow an argument, can you explain it to me please)

Define $V:=W^{\beta, 2} \subset H:=L^2$ which is compact and dense. It follows that $L^2(0,T;V) \subset L^2(0,T;H)$. Let $w^\epsilon$ be a sequence which is uniformly bounded in $L^2(0,T;V)$ and in ...
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1answer
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How do I prove the interior of subspace $\ell^1$?

Let $E:=\ell^1$ is Banach space with standard norm for $\ell^1$, $P:=\{\bar{x}\in\ell^1: \bar{x}=(x_i)=(x_1,x_2,\ldots),x_i \geq 0, \forall i \in \mathbb{N}\}$ and defined that interior of $P$ is ...
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Show the IVP has a unique solution

Assume that $f:\mathbb{R}^{n}\times \mathbb{R} \to \mathbb{R}$ satisfies (i) there exists a constant $M>0$ such that $|f(x,t)-f(y,t)|\leq M|x-y|$ for each $x,y\in \mathbb{R}^n$ and each $t\in ...
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Lifting an element isometrically (or contractively) from a quotient of a Banach algebra

Let $A$ be a (unital) Banach algebra and let $J$ be an ideal. Let $A_r$ be a closed linear subspace of $A$. Suppose that the continuous linear bijection $A_r/(J\cap A_r)\rightarrow (A_r+J)/J$ is an ...
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M bounded if and only if $\sup\{|\langle x,y \rangle|:y\in M \}<\infty$

Let $H$ be a Hilbert space. Show that $M\subset H$ is bounded if and only if $\sup\{|\langle x,y \rangle|:y\in M\}<\infty$ for every $x\in H$ My attempt: Since $H$ is a Hilbert space any set is ...
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1answer
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Bochner Integral: Derivative

Given a Banach space $E$. Consider a continuous derivative: $$F'\in\mathcal{C}(\mathbb{R},E):\quad\int_\mathbb{R}\|F'(s)\|\mathrm{d}s<\infty$$ Then its integral computes as: ...
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When does differentiability of $g\circ f$ and $f$ resp. $g$ imply differentiablity of $g$ resp. $f$?

To me the following seems intuitively true: If $f$ is differentiable at $x$ with surjective derivative then $g$ is differentiable at $f(x)$ iff $g\circ f$ is differentiable at $x$. On the other ...
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1answer
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Bochner Integral: Primitive

Given a Banach space $E$. Consider a continuous function: $$F\in\mathcal{C}(\mathbb{R},E):\quad\int_\mathbb{R}\|F(s)\|\mathrm{d}s<\infty$$ Then it has a primitive: ...
3
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1answer
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If $\sup_{T\in \tau}|y^*(Tx)|<\infty$ then $\tau$ is bounded in $L(X,Y)$

Let $X$ be a Banach space, $Y$ a normed vector space and $\tau\subset L(X,Y)$. Show that if $\sup_{T\in \tau}|y^*(Tx)|<\infty$ for all $x\in X,y^*\in Y^*$ then $\tau$ is bounded in $L(X,Y)$. ...
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Is any closed and bounded subset of a reflexive Banach space compact in the weak topology?

It seems to me that Alaoglu's theorem implies that any closed and bounded subset of a reflexive Banach space is compact in the weak topology. Is convexity of the set also needed?
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Why doesn't Alaoglu's theorem imply that $X^{*}$ is locally compact in the weak* topology?

I must be missing something basic and simple: If $X$ is a normed vector space and the closed unit ball in $X^{*}$ is weak* compact, and translations and dilations are homeomorphisms, why isn't $X^{*}$ ...
2
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2answers
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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
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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 ...