Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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3
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26 views

Compactness of a set of bounded functions in the uniform norm

Let $T$ be a non-degenerate compact interval in $\mathbb R$ and $f:\mathbb R^2\to\mathbb R$ a strictly concave function such that (a) $f(0,0)=0$, (b) $f$ strictly increases in the first argument, and ...
0
votes
1answer
9 views

Dual operator of an isometry

If $X,Y$ are Banach spaces and $\phi:X\to Y$ is an isometry, show that $\phi^*$ is surjective. I can use the equality $^\perp(ran \phi^*) = \ker\phi=\{0\}$, and also use the fact that $ran \phi^*$ ...
0
votes
1answer
13 views

Continuity of operators defined via inner products.

Let $H$ be an (in general infinite dimensional) separable Hilbert space with scalar product $<\cdot,\cdot>$. Given another inner product $<\cdot,\cdot>_2$ defined everywhere on $H \times ...
0
votes
3answers
31 views

motivation of definition of semigroup

I knew the definition of a group and semigroup. However, I do not see the point why we need the definition of semigroups without "good" algebraic properties as groups. Can someone motivate the ...
2
votes
1answer
46 views

show that every compact operator on Banach space is a norm-limit of finite rank operators (in a particular way, under the given hypotheses)

Let $X$ be a Banach space and suppose there is a net $\{F_i\}$ of finite-rank operators on $X$ such that $(a)$ $\sup_i||F_i||<\infty$ ; $(b)$ $||F_ix-x||\to 0$ for all $x$ in $X$. Show that if $A$ ...
0
votes
0answers
8 views

Showing $\sup_{\|b\|=1} \|ab\| = \sup_{\|b\|=1} \|b^\ast ab\|$

I wanted to show $\displaystyle \sup_{\|b\|=1} \|ab\| = \sup_{\|b\|=1} \|b^\ast ab\|$. (I also showed $\|a\|=\sup_{\|b\|=1}\|ab\|$.) One direction was easy: For all $a,b \in A$: $$ \|b^\ast a b \| ...
0
votes
1answer
21 views

Let $T:X\to Y$ be a linear operator and $dim X=dim Y=n<\infty$. Show that ${\scr{R}}(T)=Y$ if and only if $T^{-1}$ does not exist.

Let $T:X\to Y$ be a linear operator and $\dim X=\dim Y=n<\infty$. Show that ${\scr{R}}(T)=Y$ if and only if $T^{-1}$ exists. I'm not sure how to start this one. Any help would be nice.
0
votes
0answers
15 views

when does bijective map exist for any pair of rational function?

Let me ask kind of different questions than former ones. Given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}\text{, and }\frac{P_3(y_1,y_2,\dots,y_n)}{P_4(y_1,y_2,\dots,y_n)}$$ where $P_i$ ...
1
vote
1answer
18 views

Prove that Hölder condition in $\Bbb R^n$ implies continuity

$f:I\subset \Bbb R^n \rightarrow \Bbb R^m$ is said to be Hölder continuous if $\exists$ $\alpha>0$ and $M>0$ such that $\|{f(x)-f(y)}\| \leq M\|x-y\|^\alpha$, $ \forall x,y \in I$, ...
1
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0answers
28 views

Is Sobolev regularity propagated under evolution?

Given a well-posed initial problem in a domain $\Omega$ of the form: \begin{equation} \square\phi=f \end{equation} where $\square$ is the wave operator, $f\in L^{2}(\Omega)$, with initial ...
0
votes
1answer
20 views

How do I show that $\sup_{0\leq x\leq1}|g(x)| \geq \int_0^1|g(x)|dx$

I want to show that $\sup_{0\leq x\leq1}|g(x)| \geq \int_0^1|g(x)|dx$ for real-valued $g$ that is continuous for $0\leq x\leq1$ . Is it enough to say that $|g(x)| \leq \sup_{0\leq x\leq1}|g(x)|$ ...
0
votes
0answers
6 views

extension of semilinear functional in cone.

I'm studying Nigel Kalton's work in extrapolation Banach space theory (paper: Differentials of complex interpolation processes for Kothe function spaces). My question is: Let $T$ be a cone contained ...
1
vote
0answers
20 views

“Almost” Hilbert spaces

This question is a bit (very?) vague. Is there some notion of how "close" a Banach space is to being a Hilbert space? What I have in mind is something like a real or complex valued function on ...
0
votes
2answers
62 views
+50

Any relations between the weak topology on a Banach Space and the weak topology on CW complexes?

I'm learning about CW complexes, which we'll say are topological spaces $X$ that admit a filtration $\emptyset \subset X^0 \subset X^1 \subset \cdots \subset X^n \subset \cdots$, with $X=\bigcup X^n$, ...
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0answers
52 views

Qualitative properties of solutions to a ordinary differential equation.

I have this problem : $$\begin{cases} -(p(t)u'(t))'=f(t,u(t))\\u(0)=u(+\infty)=0\end{cases}$$ we have that $u$ is continues, $f:\mathbb{R}^+\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous and ...
3
votes
1answer
59 views

Relative countable weak$^{\ast}$ compactness and sequences

I am finding serious difficulties in understanding some things about relative countable compactness and the use of sequences in proving it by my functional analysis text, Kolmogorov-Fomin's. For ...
0
votes
1answer
31 views

Show that R is an equivalence relation on X for x, y in X iff f(x) = f(y)

$f:X→Y$ $x,y ∈ X,xRy$ iff $f(x) = f(y)$ Show that R is an equivalence relation on X. Also when $X = Y = \mathbb{R}$ and $f: \mathbb{R} \to \mathbb{R} $ with $x \mapsto x^2$ for all $x∈R$ find the ...
0
votes
0answers
19 views

How to Find Frame with lower dimension for $C^n$

Let ${f_k}$ be a frame for $C^n$ with unit norm and frame lower bound A>1. Let I(index set) be subset of {1,2,...,m} such that $|I|<A$, where $m$ is the dimension of the frame. then ${f_k}$ where ...
4
votes
0answers
131 views
+50

Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
2
votes
1answer
60 views

Modulus of continuity properties and uniform continuity.

Let $f:[a,b]\rightarrow \mathbb{R}$ bounded and $\omega(f,r)=\sup\{|f(x)-f(y)| \colon x,y \in [a,b], \ |x-y|<r\}$ (called modulus of continuity of $f$ EDIT note: the original question is in ...
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votes
0answers
16 views

A Problem from Seminorm [on hold]

Let $X$ and $Y$ be two topological vector spaces and $f: X \rightarrow Y$ be a linear map.. Show that for every continuous seminorm $p$ in $Y$ if $p\circ f$ is a continuous seminorm in $X$, then $f$ ...
-1
votes
1answer
20 views

An Open Mapping Problem [duplicate]

Consider two topological vector spaces $X$ and $Y$ where $Y$ is finite dimensional. Let $f:X \rightarrow Y$ be a surjective linear map. Prove that $f$ is an open mapping.
1
vote
1answer
25 views

A question on the 2-norm defined by $||x||_2=\sqrt{\sum\limits_{i=1}^n|x_i|^2}$

A question on the 2-norm defined by $||x||_2=\sqrt{\sum\limits_{i=1}^n|x_i|^2}$ I am trying to prove the triangle inequality of this norm. So far I have that: \begin{align} ...
4
votes
1answer
73 views

Greatest open ball of invertible elements in a Banach algebra

Let $a$ be an invertible element of a Banach algebra $A$. Then we know that also each $a+b$ with $b\in A$ and $||b||<||a^{-1}||^{-1}$ is invertible. Now my question is whether ...
0
votes
1answer
24 views

Measurability of Modulus

Context: This problem came up while reading an essay on Bochner integrability. Let $\Omega$ be a measure space and $E$ a Banach space. Consider two plain functions $f:\Omega\to E$ and $g:\Omega\to ...
2
votes
1answer
29 views

The image of Banach space under its embedding provided by the Banach-Mazur theorem

It is a very nice argument of Banach and Mazur which they use to show that every Banach space $X$ is isometric to a subspace of the space $C(B_{X^*})$, where $B_{X^*}$ is the unit ball of the dual ...
2
votes
1answer
25 views

The uniform homeomorphism between $\mathrm{Prob}(\Gamma)$ and $l^{2}(\Gamma)$

Here is a quotation of book "C*-algebras and Finite-Dimensional Approximations" by Nate and Taka. In the proof of Proposition 4.4.5. (In P132), the author says: The assertion $(1)\Longleftrightarrow ...
2
votes
0answers
16 views

Show that the set $\{e^{\pm i (n-1/4)t}: n=\pm 1,\pm2,\pm3,\ldots\}$ is not a basis for $L^2[\pi,\pi]$

Show that the set $\{e^{\pm i (n-1/4)t}: n=\pm 1,\pm2,\pm3,\ldots\}$ is not a basis for $L^2[\pi,\pi]$. (HINT: The series $$\sum_n c_n e^{i\lambda_n t}$$ with $\lambda_n=n-1/4$, diverges in ...
3
votes
3answers
63 views

Compactness of $L^p$ inclusion into $L^q$

Let $ i \colon L^p (0, 1) \longrightarrow L^q (0, 1) $, when $ p \ge q $ the canonical inclusion. It is clearly continuos but never compact: I cannot succeed in showing the last point. Thanks for the ...
0
votes
0answers
12 views

Clues in looking into extreme points of 2x2 matrices [on hold]

Could anyone give me a clue as to where to start with finding the extreme points of matrices? I have looked at extreme points of $\mathbb{R}^n$ and things but not really sure where to begin! Any ...
0
votes
1answer
16 views

how to prove a uniformly convex Banach space is reflexive

How to prove a uniformly convex Banach space is reflexive. This is an amazing result, it will be easier to prove that a Banach space is reflexive. I only know that there is not much relation between ...
1
vote
2answers
34 views

Schauder Basis Confusion

I'm learning out of the Kreyszig book for Introductory Functional Analysis and I'm having trouble understanding one of the questions. From section 2.3, question 10 reads: Show that if a normed space ...
1
vote
2answers
40 views

Orthogonal basis in infinite dimensional spaces

It is a well known fact that a symmetric bilinear form $g$ on a finite-dimensional vector space $V$ over a field $F$ of characteristic $0$ admits a orthogonal basis $\{e_i\}$, i.e. $\{e_i\}$ is a ...
1
vote
0answers
14 views

The use of Schauder fixed point in ladyzehskaya

The book Linear and Quasilinear equations of parabolic type gives the uniform parabolic pde theory in the literature. Ladyzhenskaya use Leray-Schauder rather than Schauder fixed point theorem. why? ...
2
votes
1answer
61 views

Is it true that $L^2$ is compactly embedded in $(W^{1,2}_{0})^{\ast}$?

Is it true that $L^{2}(\mathbb R^{n})$ is compactly embedded in $(W^{1,2}_{0}(\mathbb R^{n}))^{\ast}$? If so, how can I prove it? Context I've just started to study Functional Analysis. I tried to ...
2
votes
1answer
24 views

Supremum of integrals of products of functions in $L^p$ space

Here is the problem I'm dealing with I'm not having success with...well, anything. Any hits on how I could get started and where I would go? edit: information on $L^p$ space
2
votes
1answer
20 views

An inequality using Sobolev norms

Let $\| \cdot \|_{H^s(\mathbb R)}$ be the usual Sobolev norm in $\mathbb R$ and $r>0$. If we have $$ \|f\|_{L^\infty(\mathbb R)} \le \| f\|_{H^k(\mathbb R)} $$ for all $k>r$, then the inequality ...
2
votes
1answer
2k views

The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms. [duplicate]

If we are in a sequence space, then the $ l^{p} $-norm of the sequence $ \mathbf{x} = (x_{i})_{i \in \mathbb{N}} $ is $ \displaystyle \left( \sum_{i=1}^{\infty} |x_{i}|^{p} \right)^{1/p} $. The $ ...
3
votes
0answers
49 views
+50

Question about proof of Browder, Minty Theorem

Could someone please assist with the following question: In the following set of notes, I am interested to know how the author obtains "By Lemma 1.11, the Galerkin equations (2.5 has a solution ...
0
votes
2answers
60 views

How does the max of $\prod_i a_i$ work?

Here are two succinct statements of the 'same' question: Statement 1: Take $a>0$ and $S \subseteq \mathbb{R}^N; S=\{(x_1,\dots,x_N)| \frac{1}{N}\sum_i x_i = a; x_i>0\}$. Define a 'product ...
5
votes
0answers
40 views

Can any two disjoint nonempty convex sets in a vector space be separated by a hyperplane?

Let $V$ be a normed vector space over $\mathbb{R}$, and let $A$ and $B$ be two disjoint nonempty convex subsets of $V$. A geometric form of Hahn-Banach Theorem states that $A$ and $B$ can be separated ...
2
votes
1answer
27 views

Inner product in Besicovitch space

Besicovitch space is a space constructed in the following way: We take the closure (with respect to the uniform convergence topology) of a linear span: ...
0
votes
0answers
16 views

Comparing notions of continuity

I have trouble distinguishing 3 different types of continuity Uniform continuous Sequential continuous Equicontinuous Could someone explain the difference between 3 and give some examples? I am ...
0
votes
1answer
23 views

Prove that solution of a variational problem exists

Let $X$ is a Hilbert Spaces We define two operators $$a:X\times X\rightarrow\mathbb R$$ and $$b:X\rightarrow\mathbb R$$ where $a$ is a symmetric, bounded, strongly positive operator, and $b$ is a ...
0
votes
1answer
38 views

Are all definite integrals considered functionals?

In my Optimization class, we are messing around with some Calculus of Variations in an effort to find functions which minimize functionals. In these cases, the spaces we're working with are spaces ...
1
vote
0answers
22 views

Weak separability.

How I can show the following statement? Let $E$ be a normed space and $A\subseteq E$. Then $A$ is separable if and only if $A$ is weak-separable. If $A$ is separable, is clear that $A$ is ...
0
votes
0answers
20 views

boundary conditions and existence theorem

I am studying existence and uniqueness of the weeks elution of a system of nonlinear parabolic PDE subject to initial and boundary conditions. I wonder whether changing boundary conditions will lead ...
0
votes
1answer
9 views

Is there any metric $d$ of $\mathbb R^n$, $n<\infty$ such that $\mathbb R^n$ is bicompact and no norm induces $d$

There are some simple metrics can't yielded by norm .But add bicompact,I can't structure such example. In fact ,I want to know the condition of metric can be yielded by norm. Sorry for my poor ...
2
votes
1answer
28 views

Spectrum of integral operator

Given $g\in C^1([0,1]\times[0,1])$, consider the operator $$Tu(x) = \int_0^1 g(x,t) u(t) dt$$ defined on $u\in C([0,1])$. Discuss the spectrum of T. My attempt: First I can show that $T$ is ...
3
votes
1answer
21 views

Operator norm and Hilbert Schmidt norm

I'm looking for a proof of \begin{equation} ||T||\leq ||T||_{HS}, \end{equation} for which it is sufficient to show \begin{equation} ||Tx|| \leq ||x|| \cdot ||T||_{HS} \forall x\in H, x\not=0 ...