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

learn more… | top users | synonyms

0
votes
2answers
26 views

$L^p$ spaces and proper inclusion

Let $1≤p < q$. Prove that $L^p(\mathbb{R}) \subset L^q(\mathbb{R})$ and the inclusion is proper. I am unsure how to begin this or even prove it about $L^p$ spaces and Banach spaces.
0
votes
1answer
18 views

Isomorphism, Separable spaces

I am trying to show that: If there are sequences $(x_n)\subset X$ and $(y_n)\subset Y$ where $X$ and $Y$ are separable Banach spaces, such that $\overline{sp}\{x_n \mid n\in\mathbb{N}\}=X$ and ...
1
vote
1answer
14 views

Analytic sets,, effros borel structure

Let SB denote the set of closed subspaces of $C(2^\mathbb{N})$ equipped with the Effros Borel structure, and $A\subset$ SB be analytic. I am reading a proof that says $A_\sim = \{Z\in $SB $ \mid $ ...
1
vote
1answer
25 views

Closed set in Baire space

I am reading a book on Banach spaces. It introduces the Baire space $\mathcal{N}=\mathbb{N}^\mathbb{N}$ as the product of infinitely many copies of $\mathbb{N}$ with the discrete topology. We have ...
1
vote
1answer
27 views

Is the relative interior of a subspace which is not closed empty?

In a general Banach space, the relative interior of a linear subspace which is not closed is empty, why ?
1
vote
1answer
18 views

Does every closed subspace of a dual space correspond to a closed subspace of its predual?

Suppose $X$ is a Banach space with dual space $X^*$. If $Y$ is a closed subspace of $X$, then $Y^\perp=\{x^*\in X^*: x^*(y)=0 \text{ for all } y\in Y\}$ is a closed subspace in $X^*$. I am wondering ...
0
votes
1answer
19 views

Strong convergence of product of operators on a Banach space

If $\{T_n\},\{S_n\}$ are two sequences of bounded operators on a Banach space $X$, such that $\{T_n\}$ converges weakly to $T$, and $\{S_n\}$ converges strongly to $S$, does it follow that $T_nS_n\to ...
0
votes
1answer
18 views

Convergence in the weak operator topology implies uniform boundedness in the norm topology?

If $\{T_n\}$ is a sequence of bounded operators on the Banach space $X$ which converge in the weak operator topology, could someone help me see why it is uniformly bounded in the norm topology? I ...
4
votes
1answer
55 views

Categorical Banach space theory

Consider the category $\mathsf{NormVect}_1$ of normed vector spaces with short linear maps$^{\dagger}$ and the full subcategory $\mathsf{Ban}_1$ of Banach spaces with short linear maps. Both ...
1
vote
1answer
43 views

Orthogonal in inner product space

Let $(X,<.>)$ is an inner product space prove that $x$ and $y$ are orthogonal if and only if $||x+αy|| \ge ||x||$ for any scalar $α$ . The first direction if $x$ and $y$ are orthogonal ...
0
votes
2answers
44 views

Prove or disprove that T:[0,2π] -> [0,2π] given by Tx = sin(2014x) is a contraction

i know that if we assume $T:[a,b] \to [a,b] $ and if $|T'(x)| ≤ α \space \forall \space a≤x≤b$ then T is a contraction . but unsure of how to apply that to this question
1
vote
0answers
42 views

dual ball of L^1 w*-sequentially compact?

Where could I find a direct proof showing that the dual ball of L^1 is w*-sequentially compact? Since $(L^1)^*=L^\infty$, I mean the unit ball $B_{L^\infty}$ with the topology $\sigma(L^\infty,L^1)$. ...
2
votes
2answers
54 views

Banach space it isn't Hilbert space [duplicate]

How can give me two or three examples about Banach spaces which it is not Hilbert spaces with proof ( I mean why it isn't Hilbert spaces ) ?
0
votes
1answer
27 views

Coanalytic families of Banach spaces

Is it true that if $G$ is a coanalytic family of separable Banach spaces, which is not Borel, and $H\subset G$ is not Borel, then H is coanalytic? This is something I have come across. I am reading a ...
0
votes
0answers
12 views

Tensor norm for matrix algebra over an $L^p$ operator algebra

This is a question about whether a certain tensor norm has a certain property. The setting is that of $L^p$ operator algebras (i.e. norm-closed subalgebras of $L^p(X,\mu)$ for some measure space ...
0
votes
0answers
21 views

if $f$ is in Banach space, then $\nabla f $ is in the dual space?

I am not very deep in advanced real analysis. Could you help me decipher the following two phrases hold? 1) if $f$ is in Banach space $\mathcal{B}$, then $\nabla f $ is in the dual space ...
0
votes
0answers
27 views

Krein Milman Property

If a closed bounded (not compact) set $X$ in a Banach space $B$ (like $L^1$) has extreme point(s), must the max of a linear functional defined on $X$ occur at one of them? I suppose it depends on $B$. ...
3
votes
1answer
26 views

Multiplicative linear functionals on subalgebras

If $A$ is a commutative $C^\ast$ algebra and $C$ is a $C^\ast$ sub algebra of $A$ is it true that the characters on $C$ are just restrictions of characters on $A$. The reason I am asking this because ...
0
votes
0answers
29 views

$d(x,L)=\max\{f(x) \,| \, f\in L^{\perp},\, \|f\|=1\}$

Let $X$ be a normed space and $L$ its subspace. Let $L^{\perp}$ be a set of all functional of whose kernel contains $L$. Then $d(x_0,L)=\max\{f(x_0) \,| \, f\in L^{\perp},\, \|f\|=1\}$ I read a ...
1
vote
0answers
26 views

the second derivative for a composite function

Let $f: U\rightarrow \mathbb{R}$ where $U\subseteq X$ a normed vector space and suppose $f(\mathbf{u})\in C^{2}$.Let $\mathbf{x}\in U$ and let $r>0$ be such that $B(\mathbf{x},r)\subseteq U. $ ...
0
votes
1answer
36 views

How to prove a space is a dual space?

How does one go about proving that a space is a dual space? The only thing I can think of is to prove that the space is isomorphic to a dual space. Is there a better way to do this? Thank you
1
vote
1answer
40 views

A paradox derived from the open mapping theorem

The problem comes from Erwin Kreyszig's Introductory Functional Analysis with Applications, section 7.4, problem 4: Let $T:l^2\mapsto l^2$ be defined by $y=Tx, x=(\xi_j), y=(\eta_j), ...
4
votes
0answers
50 views

Proving completeness of $L^p$

I want to make sure my understanding of the proof is correct. For a Cauchy sequence $\{f_n\}$ in $L^p$, we want to find a $f\in L^p$ such that $f_n\stackrel{L^p}\to f$ Now, skipping the ...
1
vote
1answer
24 views

A question about regulated functions

Let $X$ be a Banach space and consider the following definition. Definition. $f:[a,b]\to X$ is regulated if it has one-sided limits at every point of $[a,b]$, i.e. for every $c\in [a,b)$ there is a ...
4
votes
0answers
67 views

Are “most” operators on an infinite-dimensional complex Banach space “diagonalizable”? [closed]

This is true for finite-dimensional spaces, of course. To be precise, let $T$ be an operator on a complex Banach space $X$ which is not finite-dimensional. For each $\lambda \in \mathbb{C}$, let ...
5
votes
0answers
43 views

Isomorphism on dense subset

I am wondering if the following could be done. I want to show two Banach spaces $X$ and $Y$ are isomorphic. If $A$ is dense in $X$, and $B$ is dense in $Y$, is it sufficient to show there is an ...
1
vote
1answer
35 views

Separable spaces isomorphic

I am reading a proof and it states the following without proof: Two separable Banach spaces $X$ and $Y$ are isomorphic iff there are sequences $(x_n)\subset X$ and $(y_n)\subset Y$ such that ...
2
votes
2answers
35 views

Showing that $L^2\subset L^1$ for $L^2([0,t_f])$, with $t_f$ a fixed positive number.

I saw demonstrations using the Cauchy-Schwartz Inequality but I am still not convinced because the Inequality is as follows : $$ \left |\langle f,g\rangle\right | \leq \left \|f \right \|_{L_2} . ...
1
vote
2answers
48 views

Examples of Sobolev Spaces

I would like to apologize in advance for this trivial question! Does constant functions $u \equiv C$ and, in partucular, $u \equiv 0$ belong to $W_{0}^{1,2}(\Omega)$? Update 2: $\Omega \subset ...
1
vote
1answer
57 views

closed, convex, absorbing subset of a banach space

There is a nice theorem that every closed, convex, absorbing subset of a banach space includes an open ball arround $0$. Can you give an example where the theorem fails if we do not assume the subset ...
1
vote
1answer
34 views

spectral theory expandable to arbitrary polynomials?

Given a Banach space $X$ and closed operators $A_i$ ($i \in \left\{0,...,n\right\}$) which have a common domain $D$ that is dense in $X$. An obvious candidate for the title of "generalised resolvent ...
7
votes
1answer
157 views

Does Continuity in Weak Operator Topology imply Continuity in Strong Operator Topology?

This homework problem has puzzled me for almost a year. As nobody in the class has figured it out, I would like to seek a proof or a disproof here. Problem (Prove or Disprove) Let $X$ be a Banach ...
1
vote
0answers
31 views

Are continuous bounded functions a subspace of $L^2$?

I have a problem where I need to work with functions that are square-integrable, bounded and continuous, i.e. the space $ L^2 \supset X = \left\{ f \in L^2 \mid f \text{ bounded, continuous}\right\} ...
0
votes
0answers
30 views

(Riesz's lemma) A closed subspace of a Banach space

Let V be a Banach space over R Let W be a proper closed subspace of V Prove : For any $\epsilon > 0$, there is a v $\in$_V_ such that ||v||=1 and ||v+w||$\geq$ $1$ - $\epsilon$ And my proof ...
1
vote
1answer
29 views

Banach space and its closed subspace. a vectors satisfying inequality.

V=a Banach space over R W=a proper closed subspace of V Prove : For any $\epsilon > 0$, there is a v $\in$_V_ such that ||v||=1 and ||v+w||$\geq$ $1$ - $\epsilon$ I have shown that there exists ...
1
vote
1answer
21 views

Existence of a vector whose norm is 1 in a Banach space

Given a finite dimensional Banach space V over reals, I have to show that there exists $v \in V$ such that $\|v\|=1$ At first, I thought that there's an identity element I in V. And $\|I\|=1$. But ...
-1
votes
0answers
16 views

When is a norm of identity one?

Is there a specific condition that makes a norm (any norm) of identity equal to one in any Banach spaces? Thanks.
1
vote
1answer
19 views

Projection on closed subspace of $L^1$, $L^{\infty}$

For $p=1,\infty$ let $K$ be a closed subspace of $L^p(\mathbb{R},m)$. According to this question, it should be easy to find examples of $K$ and $f\in L^p(\mathbb{R},m)$ such that there exists a ...
2
votes
1answer
45 views

Projection on closed subspace of $L^p$, $1<p<\infty$

Let $1<p<\infty$ and $K$ be a closed subspace of $L^p(X, \mathcal{M}, \mu)$. If $f\in L^p$ then there exists a unique $h\in K$ such that $||f-h||_p$ equals $$ \text{dist}(f,K)=\inf_{g\in ...
-1
votes
0answers
24 views

Closed hyperplanes of a Banach space are isomorphic

Let $X$ be a Banach space and $S_1$ and $S_2$ two of its closed subspaces of co-dimension 1. Prove that they are isomorphic (i.e. there is a bounded bijective linear operator between them).
2
votes
1answer
34 views

Is this a Hilbert space?

For $n\geq 2$, we let $\mathcal{H}$ be the complex vector space of all complex-valued functions on $[0,1]$ such that (a) $f(0)=0$, (b) for $1\leq k\leq n-1$, $f^{(k)}$ exists everywhere and is ...
0
votes
1answer
51 views

Can the ball $B(0,r_0)$ be covered with a finite number of balls of radius $<r_0$

Consider an infinite dimensional Banach space $X$. Let $B(0,r_0)$ be the ball with radius $r_0$. We know that the ball $B(0,r_0)$ is not relatively compact, so it is not totally bounded. This implies ...
1
vote
3answers
32 views

Surjectivity of $Id-A$ for linear operator $A$ on Banach space with $\|A\|<1$

Let $X$ be Banach space and $A:X\rightarrow X$ linear opeartor such that $\|A\|<1$. It is clear that $Id-A$ is injective. Why is it also surjective?
-1
votes
0answers
28 views

Compact Operators: Adjoint

Given Banach spaces $E$ and $F$. Consider a bounded operator: $$T:E\to F:\quad\|T\|<\infty$$ As a result due to Schauder: $$T\text{ compact}\iff T'\text{ compact}$$ How to prove this fact?
2
votes
1answer
38 views

Banach space of p-Lipschitz functions

Given $p\in\mathbb{R}$, consider the space: $$ Lip(p) = \left\{f:[0,1] \longrightarrow \mathbb{R} : \mbox{ $f$ is $p$-Lipschitz} \right\}$$ i.e.: there is $M>0$ such that ...
2
votes
3answers
73 views

Infinite dimensional spaces other than functional spaces

"Functional analysis" is the study of infinite dimensional spaces equipped with inner product, norm, topology...etc. The most interesting spaces are the spaces of functions/operators and sequences. I ...
2
votes
3answers
59 views

What is abstraction of direction in considering vectors such as used in Engineering & Physics?

In the use of vectors of engineering and physics, we encounter objects that obey the axioms of a vector space but also have two new attributes of length (or, magnitude) and direction (e.g. direction ...
3
votes
1answer
34 views

is the complexification of a finitely strictly singular operator itself FSS?

Let $X$ and $Y$ be real Banach spaces, and let $X_\mathbb{C}$ and $Y_\mathbb{C}$ denote their respective complexifications. Suppose $T:X\to Y$ is a bounded linear operator which is finitely strictly ...
0
votes
1answer
32 views

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} ...
-2
votes
0answers
83 views

Prove that $(\mathbb R^n,\|\cdot\|_3)^*= (\mathbb R^n,\|\cdot\|_{1.5})$ with full details

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 ...