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, ...

learn more… | top users | synonyms (1)

0
votes
0answers
37 views

How can I prove that this function doesn't have a second weak derivative?

I'm trying to determine what weak derivatives the function $$ f(x)=\begin{cases} x&\mbox{if }0<x<1,\\ 1&\mbox{if }1\leq x<2, \end{cases} $$ has. I already managed to prove that it ...
1
vote
1answer
39 views

Orthogonal Complements of polynomials in $L^2[0,1]$

I have two very difficult questions in my home work in function analysis, in which I have two calculate the complements of the following sets, in $L^2[0,1]$: All polynomials in the variable $x^2$ ...
1
vote
1answer
19 views

equivalnce of linear functions, which one's kernel includes the other's

The following is from my homework. PLEASE don't reveal all the solution, but leave at least something for my imagination. Let $X$ be a normed space. Let $\phi,\psi : X → \mathbb C$ be linear ...
1
vote
1answer
63 views

Proving this set is dense in $\ell^2$

I found this weirdest question and was wondering how could this be proved. This question is a part of a beautiful semi-constructive built of two dense disjoint convex sets in $\ell^2$, which I find ...
0
votes
1answer
26 views

A question in functional analysis about bounded linear operator.

Suppose $Banach$ Space $E$ is the direct sum of its closed subspaces $L$、$M$, and $M$ is finite-dimensional, $T$ is a bounded linear operator from $E$ to itself. Prove that $T(E)$ is a closed subspace ...
-1
votes
1answer
19 views

Showing that the trace of a positive operator is independent of orthonormal base [on hold]

let $T$ be a positive operator on a separable Hilbert space. let $\{e_n\}$ be an orthonormal base for the space, and suppose the trace of $T$ is finite, i.e. $$tr(T)= \sum_{n=1}^{\infty}(Te_n,e_n) ...
1
vote
1answer
57 views

Example of a wot convergent net but not $\sigma -$ weak convergent

Let $B(H)$ be the space of bounded linear operators. Define the $\sigma-$ weak topology on it by seminorms $p_{h,k} (x)=|\sum_{n\geq 1}(xh_n,k_n)$ where $h=\{h_n\}\subset H ,~~ k=\{k_n\}\subset H $ ...
-2
votes
0answers
15 views

Weakly convergence w.r.t weakly closed [on hold]

If $K$ is weakly closed in a Banach space $X$,if $\{u_k\}\in K$ converges to $u$ weakly, how to prove that $u\in K$
0
votes
1answer
25 views

Uniformly Converging Sequence but Not Normly Converging

I want to find a sequence $(f_n) \subset \mathcal L^1(\mathbb R)$ of integrable functions which uniformly converges to $0$ but with $\int f_n \nrightarrow 0$. I came up with the following example: the ...
2
votes
1answer
49 views

Infinite direct sum of Hilbert spaces

Let $\{H_i\}_{i \in I}$ be an infinite collection of Hilbert spaces. I am trying to understand their "Hilbert space direct sum". $\bigoplus H_i$ (algebraic sum) is an inner product space in a ...
2
votes
1answer
27 views

Polar decomposition

Every $x\in B(H)$ has a representation such as $x= u|x|$ (by polar decomposition) where $u$ is a partial isometry. In a paper, the author claims $$u = strong - \lim_{\epsilon\to 0} ...
0
votes
2answers
40 views

$\frac{\varphi(x+h)-\varphi(x)}{h}\to\varphi'(x)$ uniformly as $h\to 0$?

Let $\varphi$ be a bounded, differentiable function on $\mathbb{R}$ such that $\varphi'$ is bounded and uniformly continuous on $\mathbb{R}$. We want to prove that ...
1
vote
1answer
31 views

The polar of a set: Importance and Applications

Given a duality $(X,Y)$, for any subset $A\subset X$, we can define the polar set $A^{\circ}\subset Y$. In this sense, the polar relates sets and sets in the dual space. Since cones are sets, we can ...
0
votes
2answers
24 views

Prove $d(x,y)=\sup _{n} \left| \sum _{k=1}^{n}(x_k-y_k)\right |$ is a metric

Let $\gamma$ be the set of convergent series.$$\gamma = \{x=(x_k), x_k \in \mathbb{R} : \sum x_k <\infty\}$$ Prove that $(\gamma , d)$ is a metric space, with $$d(x,y)=\sup _{n} \left| \sum ...
3
votes
0answers
79 views

Functional inequalities involving cubing and incrementing

Consider the set $S$ of positive increasing invertible functions $f$ satisfying: $$f((x+1)^3-1)≤(f(x)+1)^3-1$$ $$f(x^3)≥(f(x))^³$$ $$f(x)+1≤f(x+1)$$ for all positive real $x$. Clearly the identity ...
2
votes
1answer
20 views

Some closed subspace of $l_2$?

$(a)$ I was trying to define a continuous linear map $T$ on $l_2$ whose kernel would be the $A$ and can conclude $A=T^{-1}(0)$ and hence closed set? could anyone help me to solve any of one?
2
votes
1answer
43 views

Theorem 2.3-2 in _Introductory Functional Analysis With Applications_ by Erwine Kryszeg

Here's the statement of Theorem 2.3-2 in the book mentioned above: Let $(X,||\cdot||)$ be a normed space. Then there is a Banach space $\hat{X}$ and an isometry $A \colon X \to W$, where $W = A(X)$, ...
3
votes
1answer
30 views

Ultraweak closed left ideal of a von Neumann algebra

The following is a proposition of Takesaki's Operator Theory: My questions are: 1- He claims for two sided ideal $\cal m$, $e \in M\cap M'$. While I think for $\sigma -$ weakly closed two sided ...
5
votes
2answers
84 views

Example (?) of a Banach space containing an uncomplemented copy of itself

I was wondering the following: Background Question: Does there exist a Banach space $X$ which contains a copy $X_0 \subset X$ of itself that is not complemented? By "$X_0$ is a copy of $X$", I ...
0
votes
2answers
43 views

Power Series: Derivative

Given a Banach space $E$. Consider a series: $$|t|\leq R:\quad\sum_{k=0}^\infty A_k t^k\quad(A_k\in E)$$ Is there an elegant proof of: $$\left(\sum_{k=0}^\infty A_k t^k\right)'=\sum_{k=0}^\infty A_k ...
4
votes
0answers
144 views
+50

On the Hölder regularity of an integral function

Let $n\geq 3$. Let $\Omega$ be an open and bounded subset of $\mathbb{R}^n$. Let define $X_0$ as the space of functions $f:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $f(x,\cdot)$ is ...
0
votes
0answers
33 views
+50

Homology and critical groups

I have this theoreme from the paper: J. Liu, The Morse index of a saddle point, 1989 My first question is what is $\tau$ is $\tau$ a chain ? so $I_m$ is the standard simplex ? if it is this why ...
2
votes
0answers
39 views

norm over differentiable functions computable from derivatives only

I'm running an optimization algorithm, minimizing the norm $||f-\hat f||$ of some objective function $f(x_1,x_2,x_3,y_1,y_2,y_3)$. The function $f$ cannot be computed directly, but its second ...
0
votes
2answers
26 views

The fixed point in Brouwer's Theorem need not be unique.

What does it mean for a fixed point to be unique? I'm thinking that it means that you can have multiple values of a fixed point. But, a fixed point is one where $f(x) = x$. So to have repeated ...
2
votes
1answer
48 views

If $E$ is not complemented in $X$, is $E \oplus \{0\}$ not complemented in $X \oplus Y$?

Question: Let $X$ be a Banach space, and let $E \subset X$ be a closed subspace such that $E$ is not complemented in $X$. Does it follow that $E \oplus \{0\}$ is not complemented in $X \oplus Y$, ...
1
vote
1answer
14 views

Operator which is invariant in terms of left shift

Let $l:l^{\infty}(\mathbb R)\rightarrow \mathbb R$ be a linear map( where $l^{\infty}$ is the set of bounded sequences) such that the following holds: (i) $l(Tx)=l(x)$, where T is the left shift ...
0
votes
1answer
15 views

Proof (?) about weak contractions. Please check to see if I'm going about this correctly?

If $f:M \rightarrow M$ satisfies that $\forall x,y \in M$, if $x≠y$ then $d(f(x),f(y)) < d(x,y)$, then $f$ is a weak contraction. Is a weak contraction a contraction? I saw a counter example on ...
0
votes
1answer
27 views

Sobolev space $W^{1,2}((0,1))$ and boundary ODE - how does integration by parts goes?

As a part of a question about $W^{1,2}((0,1))$, I want to get a boundary ODE on $g$ and don't quite know how to integrate (?) in order to get the equation. let $g\in C^2[0,1]$ be our variable, $f\in ...
2
votes
0answers
29 views

Sampling theorem.

Let us consider \begin{equation} \hat{f}(x)=\sum_{n\in \mathbb Z}\left\langle\hat{f},e^{i n x}\right\rangle_{L^2[-\pi,\pi]} e^{i n x} \ \ \ \ \ \ \ \ (1) \end{equation} where $\langle g, ...
0
votes
2answers
26 views

Unbounded operator

Assume you have an operator $T : \operatorname{dom(T)}\rightarrow H$. Now we also know that $ran(T)$ is finite-dimensional. Does this imply that $T$ is bounded?( So is $T$ a bounded map $T \in ...
5
votes
1answer
79 views

orthonormal sequence in $L^2[0,1]$ - how to prove these following equivalent terms?

I've been asked this following very interesting question and would like some help figuring out why it is true :) Let $u_n$ be an orthonormal sequence in $L^2[0,1]$ Prove that the following are ...
0
votes
1answer
19 views

Showing that $p(x)\mapsto p'(x)$ is not a continous linear transformation

I am trying to understand a result in Rynne & Youngson: Linear Functional Analysis. Regarding continous linear transformations, the following is stated: Let $X$ and $Y$ be normed linear ...
2
votes
1answer
30 views

continuously embedding

Let $X$ and $Y$ be two normed vector spaces, with norms $||·||_X$ and $||·||_Y$ respectively. If there are constants $C_1, C_2≥0$ such that $||·||_Y \leq C_1||·||^{1/2}_X+C_2||·||^2_X$ for every $x\in ...
1
vote
1answer
36 views

If $A$ is a $*-$ Banach algebra then $\bar A^{wot} = \bar A^{weak^*}$?

If $A$ is a $*-$ subalgebra of $B(H)$, then clearly $\bar A^{weak^*}\subset \bar A^{wot}$ (wot means weak operator topology). Also on every bounded subset of $A$, two topologies equal. Now my question ...
1
vote
1answer
47 views

Tensor product of function space and vector space

In one book I found that they treat vector valued functions as tensor product of some function space and vector space. So for example $$ C(\mathbb{R},V) \simeq C(\mathbb{R})\otimes V \\ ...
1
vote
1answer
18 views

Understanding connectedness argument in proof of Analytic Fredholm Theorem

Let $X$ be a complex Banach space, and let $D \subset \mathbb{C}$ be a domain. Let $\mathcal{L}(X)$ denote the Banach space of bounded linear transformations $X \to X$. The Analytic Fredholm Theorem ...
1
vote
0answers
16 views

Equivalence of condition on function

In a book I am studying it states that a condition on a function $g$ as follows: Given the function $g: \Omega \times \mathbb{R} \mapsto \mathbb{R}$ is a Caratheodory function satisfying $$\sup_{|u| ...
0
votes
1answer
19 views

Necessary condition of optimality for functionals

Let $C(a, b)$ denote the set of all surjective and continuously differentiable functions $\alpha:[a, b] \rightarrow [a, b]$. Consider the functional on $C(a, b)$ $$ F[\alpha(t)] = \int_a^b ...
4
votes
2answers
71 views

$e_n \to 0$ weakly in $l^\infty$

Given the the sequence $(e_n)_n$ in $l^\infty$, I want to show that that $e_n$ converges weakly to $0$ in $l^\infty$, i.e. $$e_n\rightharpoonup 0 \text{ as } n\to \infty.$$ By $e_n\in l^\infty$, ...
0
votes
0answers
27 views

What is the Vapnik-Chervonenkis dimension of sigmoidal functions?

Consider the following class of functions: $F=\{f_w:R^d \rightarrow [a,b], f_w(x)=\sigma(w^Tx), \forall x\in R^d\}$, where $\sigma(\cdot)$ is a sigmoidal function (e.g. tanh, or sigmoid so it has ...
3
votes
1answer
83 views

Best approximation for a normed vector space $X$

I am self-studying functional analysis. As far as I know, a best approximation of $X$ by a closed subspace $C \subseteq X$ exists and is unique if $X$ is a Hilbert space, a uniformly convex Banach ...
0
votes
2answers
27 views

Range conditions on a linear operator

While reading though some engineering literature, I came across some logic that I found a bit strange. Mathematically, the statement might look something like this: I have a linear operator ...
2
votes
1answer
41 views

Show that a bounded linear transformation is continuous

I am not sure what this question is asking. A linear operator $T$ between normed spaces X and Y is bounded if and only if it is a continuous linear operator. But weak topology is not metrizable. I ...
1
vote
0answers
25 views

Non-Reflexive Spaces

I read the following example of non-reflexive spaces which I do not understand. Let $X:=C([0, 1])$ be the space of continuous function on $[0, 1]$. It is mentioned that the dual of this space, $X^*$, ...
1
vote
0answers
20 views

Show that $\pi(M)'' = \pi(M'')$

Let $M$ is a $*-$ subalgebra of $B(H)$. Let $\bar H$ denote the direct sum $\sum H_i$ where $\{H_i\}$ is a family of replicas of $H$. Define $$\pi :x\in B(H) \to \bar x \in B(\bar ...
0
votes
1answer
31 views

Computing a Projection Valued Measure

I've recently begun learning about Projection Valued Measure and I'm a little confused. I understand that a Projection Valued Measure is a family of orthogonal projections $P(\Lambda)$ indexed by the ...
3
votes
1answer
29 views

How to see this is an isometry

Let $X$ be a separable Banach space, and $(f_n)$ be a countable dense subset. Recall that for each $f_n$ there exists a linear functional $l_n \in X^*$ such that $\|l_n\| = 1$ and ...
0
votes
1answer
20 views

How to see injection and boundedness

Lemma. If $A$ is a bounded linear operator defined on a Hilbert space and $\|Af\| \geq c\|f\|$ and $\|A^*f\| \geq c\|f\|$ for some constant $c$. Then $A$ has a bounded inverse. In the proof of ...
0
votes
0answers
20 views

Sobolev spaces and Cauchy sequences with respect to $L^2$-norm.

Let $z^n=(u^n,w^n,\phi^n)$ be a sequence in $H=H_*^1(0,\ell)\times H_0^1(0,\ell)\times H_*^1(0,\ell)$, where $H^1(0,\ell)$ and $H_0^1(0,\ell)$ are the usual Sobolev spaces and $H_*^1(0,\ell)=\{f\in ...
0
votes
1answer
32 views

Surjective Operators

Let $X$ be a separable Banach space. Show that there is a surjective linear operator $P: L^1(\mathbb N) \to X$ and a subspace $S \subset L^1(\mathbb N)$ such that $$\|P(x)\| = \inf_{y \in S} ...