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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2
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2answers
35 views

Space of Lipschitz continuous functions is complete

Let $X$ be set of functions $f:[-1,1]\to \mathbb{C}$ such that $f(0)=0$ and there exists $\alpha>0$ such that $$ |f(t)-f(s)|\le \alpha |t-s| $$ for all $t,s\in [-1,1]$. Equip $X$ with the norm: $...
2
votes
0answers
39 views

Example of Hilbert space non isomorphic to $L2$

I'm looking for an example of a Hilbert space that can't be seen as the countable direct sum of $L^{2}(X,\mu)$ spaces nor subespaces of $L^{2}(X,\mu)$. Some idea to start? Thanks everyone.
0
votes
0answers
20 views

Non linear functional optimization under constraints

For some given positive functions $l(t)>0$ and $h(t)>0$, such that $h(t)>l(t)$, I want to solve this functional optimization problem on $a(t)$: $\min_a\int_0^T[l(t)\cdot\min(a(t),0) + h(t)\...
1
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0answers
120 views

Problem regarding continuous embeddings [duplicate]

Given the following exercise: We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \{x \in (\mathbb R_+) : f(x) \neq 0\} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$ and for all $f \in C^...
0
votes
1answer
47 views

Spectrum of $T\in \mathcal{L}(E)$, such that $T^n=I$

Let $T:E \to E$ be a bounded linear operator, $E$ infinite dimensional Banach space, such that $T^n =I$, for $n\ge2.$ Show that $\sigma(T)\subset\{-1,1\}.$ My idea is show that $\|T\|=1$ initially,...
0
votes
1answer
24 views

Sup norm of vector-valued function

If $\vec{u}$ is a real-valued vector-valued function, say $\vec{u}=(u_1,u_2,u_3)$, is the following correct? $$\|\vec{u}\|_{\infty}=\sum_{i=1}^3\sup|u_i|.$$
0
votes
1answer
42 views

Urysohn's extension theorem

Currently I am working my way through Ernest Michael's first article on continuous selections. Here, Urysohn's extension theorem is stated as follows: For a $T_1$-space, the following properties ...
1
vote
1answer
79 views

approximating an $ L^1 $ function with a function of compact support.

Can we approximate an $L^1$ function of several variables $ f(x_1, x_2,.., x_N) $ with a continuos function $ g(x_1, x_2,.., x_N) $ of compact support in sense of $ L^1 $ $\quad $ ? That is for $\...
0
votes
1answer
22 views

Bounded linear extension of an operator whose co-domain is incomplete.

Let $X$ be a normed space and $Y$ be a Banach space. The bounded linear extension theorem states that a bounded linear operator $T : M \to Y$, where $M \subseteq X$ is dense, can be extended to a ...
-8
votes
0answers
156 views

Show that map is norm preserving and determine Eigenvalues [duplicate]

Can someone of you give me a solution for this? Let $N\in \mathbb N$. a) We define the map $\mathfrak F:(\mathbb C^N, ||\cdot||_2)\to(\mathbb C^N, ||\cdot||_2)$ by $$(\mathfrak F(x))_k := \frac{...
0
votes
1answer
34 views

Is any closed ball compact in the Weak$^*$ topology $\sigma(E^*,E)$ for a Banach Space $E$?

For a Banach Space $E$, the Banach Alaoglu Bourbaki theorem asserts that the closed unit ball in $E^*$: $$B_{E^*}= \{f \in E^* \ | \ ||f|| \leq 1 \} $$ is compact in the weak$^*$ topology $\sigma(E^*...
1
vote
1answer
19 views

Is $f$ is closed equivalent to the graph of $f$ is closed when $f$ is linear

Suppose $X,Y$ are topological vector spaces, $f:X\rightarrow Y$ is a linear map, is that true that two of the following are equivalnet: 1.$f$ is closed 2.The graph of $f$ is closed. What if $X,Y$ ...
0
votes
1answer
31 views

Writing an operator $T$ defined by $(T f)(t) = \int_{-\pi}^\pi h(t − s)f(s)ds$ as $\sum_{n \in \mathbb Z} \mu_n \langle f, \varphi_n\rangle \varphi_n$

Let $h$ be a continuous function with period $2\pi$. Define $T : L_2[−\pi, \pi] \to L_2[−\pi, \pi]$ by $(T f)(t) = \int \limits _{-\pi}^\pi h(t − s)f(s)ds$. Let $\{\varphi_n(t) =\frac{1}{\sqrt{2\pi}} ...
0
votes
1answer
21 views

dom(A) is a Banach space w.r.t. the Graph-norm

Let $X$ and $Y$ be Banach spaces and let $A:dom(A)\to Y$ be a linear operator, defined on a linear subspace $dom(A)\subset X $. Prof that the graph of $A$ is a closed subspace of $X\times Y$ if and ...
1
vote
1answer
28 views

Prove that exists a linear continuous functional satisfying…

Let $E$ be a normed space over the field of real numbers. I have to prove that given two convex sets $A$, $B$ in $E$, with positive distance between then, there exists a linear continuous functional ...
0
votes
0answers
22 views

weighted shift operator for complex Hilbert space

I am trying to solve that if H is a complex Hilbert space with orthonormal basis $\{e_n\}_{n=1}^{\infty}$ and let $\{a_n\}_{n=1}^{\infty}$ be a sequence with $\lim_{n\rightarrow}a_n = 0$. Define the ...
0
votes
1answer
19 views

Inverse Operator Theorem, counter example

Let $X=Y=C([0,1])$ be the space of continuous functions $f:[0,1]\to \mathbb{R}$. The normed space $(X,\Vert \cdot \Vert_X) $ with $\Vert f\Vert_X:=\sup_{0\leq t\leq 1}\vert f(t)\vert$ is complete and ...
4
votes
0answers
53 views

What are “generalized bases” really called, and where can I learn more?

(Notation: $f \diamond g$ means the composite $g \circ f$.) The following situation occurs frequently: We have an $\mathbb{R}$-algebra $A$, together with a distinguished set $I$ (the "indexing set"),...
1
vote
2answers
74 views

Integrating a Linear Operator $A:H\longrightarrow H$ (Matrix)

I am trying to prove a functional analysis proposition, but I got stuck. I have to integrate a matrix. In my proof I use the following matrix: Let $A$ be a self-adjoint matrix on $H=\mathbb{C}^n$ ...
1
vote
1answer
21 views

Proving two stubborn inequalities for completely positive maps in C*-algebras

I recently came across this in my studies of functional analysis in C* algebras which got me stuck: For a completely positive map between C* algebras $ \phi : A \to B $ we are to prove these two ...
1
vote
1answer
12 views

The finite intersection of absorbing set is absorbing

I got stuck proving the finite intersection of absorbing set is absorbing. Can anyone help?
4
votes
1answer
26 views

Non-self Adjoint Operator Algebra References

The problem I am working on has led me to define a norm closed sub-algebra $\mathscr{A}$ of $\mathscr{B}(\mathscr{H})$. The algebra is generated by some mild relations, and in general, will not be ...
1
vote
1answer
26 views

Finding set of functions

$ f\left(u,v\right)=u^{2}+3v^{2} $ $g\left(x,y\right)=\begin{pmatrix} e^{x}cosy \\ e^{x}siny \end{pmatrix} $ How do I determine sets of $f\left(\mathbb R ^{2} \right)$ and set of $g\left(\mathbb ...
0
votes
1answer
35 views

When different metrics which induce the same topology have the same result

For those important theorems in functional analysis, e.x. Banach-Steinhaus theorem, in the proof, we use the language of metric, but the result can be applied to any metric with the same topology on ...
0
votes
0answers
38 views

Injective Integral Operator on $L^2[0,1]$ or $C[0,1]$?

Consider an arbitrary $f \in L^2 [0,1]^+ $ where $L^2[0,1]^+$ is the function space of square integrable non negative functions. We say $T$ is an Integral Operator if $T$ is of the form , $$ T(f) = ...
1
vote
0answers
71 views

Calculate the trace of $LBB^*$, where $L:H→H$ and $B:=ΦT^{1/2}$ for some $Φ:U_0→H$, an embedding $ι:U_0→V$ and $T:=ιι^*$

Let$^1$ $U$, $V$ and $H$ be $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\...
2
votes
0answers
39 views

absolutely convergence in $l^2(\mathbb{N})$ space

Let $x=(x_i)_{i=1}^{\infty}$ be a sequence such that for all $y=(y_i)_{i=1}^{\infty}$ $\in l^2(\mathbb{N})$ $\sum\limits_{k=0}^{\infty}|x_iy_i|< \infty $ Show that $x \in l^2(\mathbb{N})$ . Can ...
0
votes
1answer
35 views

Spectrum of two Hilbert spaces

Let $H_1$ and $H_2$ be two Hilbert spaces and $U \in B(H_1,H_2)$ be unitary. Assume that $A\in B(H_2)$ and $B \in B(H_1)$ satisfy $UB = AU$. How can I prove that $sp(A) = sp(B)$ and $sp_p(A) = sp_p(B)?...
0
votes
1answer
7 views

Prove that the conjugate transpose is bijective too

Let $T:\mathcal{V}\to\mathcal{W}$ be a linea map and let be bijective. How can I prove now that his conjugate transpose, $A^*$ is bijective too? My idea was that if $T$ is bijective then there exists ...
1
vote
2answers
44 views

Unbounded operators in Hilbert space.

If $A\subset B $, $A$ not equals to $B$, are unbounded operators, I need to prove that: $(a)$ If $B$ is selfadjoint, then $A$ is symmetric, but not selfadjoint. $(b)$ If $A$ is selfadjoint, then $B$ ...
0
votes
1answer
34 views

Find functions $F(\mathbf{x})$ invariant under a map $\mathbf{x} \to \mathbf{x'}$

We introduce a map $\mathbf{x} \to \mathbf{x'}$, defined as (for example on $\mathbb{R}^3$): $$x'=f(x,y,z) \\ y'=g(x,y,z) \\ z'=h(x,y,z)$$ Note that $f,g,h$ are not all linear (or at least, I'm not ...
0
votes
0answers
15 views

Wave Equation: Distribution which maximises Entropy

Given the wave equation: $\left(\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}\right)f(t,x)=0$ and expanding $f(t,x)$ through a Fourier Transform: $f(t,x)=\int d\omega dk F(\omega,k)e^...
1
vote
2answers
31 views

Closure of Injective and Surjective Sets in $C([0,1]\rightarrow [0,1])$

Consider the space of all continuous functions from $[0,1]$ to $[0,1]$ equipped with the $\sup$ metric. Is the set of all injective functions in this space closed? What about the set of all surjective ...
0
votes
0answers
25 views

Eigenvalues of Finite Type

I want to show that the following holds: Let $T:X\rightarrow Y$ and $S:Y\rightarrow X$ be operators acting between Banach spaces. Assume that $\mu \not=0$ is an eigenvalue of finte type of $ST$. ...
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vote
0answers
24 views

Prob. 4, Sec. 4.3 in Kreyszig's functional fnalysis text: Application of the Hahn Banach Theorem

Let $X$ be a real or complex vector space, and let $p \colon X \to \mathbb{R}$ be a real-valued function satisfying $$p(x+y) \leq p(x) + p(y) \ \mbox{ for all } \ x, y \in X$$ and $$p(\alpha x) = \...
0
votes
0answers
26 views

Uniform Boundedness hypothesis

The hypothesis includes stating that the linear operators T, from X to Y, in the family of operators are bounded; how is the statement that follows in the hypothesis any different? " If for all x in ...
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vote
0answers
17 views

Is K(H) separable?

Let $H$ be an inifinite dimensional separable K-Hilbert space with $K$ could be $\mathbb{R}$ or $\mathbb{C}$. Are the compact operators $K(H)$ separable? It's well known that $\overline{F(H)}=K(H)$, ...
0
votes
1answer
90 views

Why is the dual cone of $l^1$ is $l^\infty$?

I just noticed somewhere in Convex Optimization that the dual cone of $l^1$ is $l^\infty$! (A diamond in $\mathbb{R}^2$ for $l^1$ is a square in $\mathbb{R}^2$ for $l^\infty$.) In fact I cannot ...
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votes
0answers
30 views

Hilbert Spaces and Hamel Basis

Let $H$ be a Hilbert Space of infinite dimension, $S$ a not finite orthonormal basis and $B$ a Hamel basis to $H$. i) How to show that the cardinality of $B$ is greater than or equal to the ...
1
vote
1answer
37 views

Non-commutaive Gelfand-Naimark theorem and dimension of Hilbert space

It is well known that using non-commutative Gelfand-Naimark theorem for finite dimensional $C^∗$-algebra we can obtain isometric representation on finite dimensional Hilbert space. My question is : ...
0
votes
0answers
14 views

closedness of a graph of an unbounded operator between two Banach spaces

Let T be a linear operator from a subset $D(T)$ of a Banach space $X$ to another Banach space $Y$. T is closed if a sequence $x_n$ in $D(T)$ converges to $x \in X$ and the sequence $Ax_n$ converges to ...
0
votes
2answers
31 views

The remark of closed graph theorem in Rudin's book

In Rudin's functional analysis, the remark of closed graph theorem says the statement: If $\{x_n\}$ is a sequence in $X$ such that $x=\displaystyle\lim_{n\rightarrow\infty} x_n$, $y=\displaystyle\...
0
votes
1answer
39 views

Determine and sketch the image of function?

I have to sketch the image of function $h\left(\mathbb R^{2} \right)$ as 1. a set and as 2. a geometric object. $h\left(r,\phi\right)=\begin{pmatrix} rcos\phi \\ rsin\phi \\ r \end{pmatrix} $ I ...
1
vote
2answers
66 views

Finding all possible values of a Function

Let a function be defined as $f:N\to N$ and $x-f(x)= 19\left[\dfrac{x}{19}\right] - 90\left[\dfrac{f(x)}{90}\right] \forall x\in \Bbb N$ and $1900<f(1990)<2000$. Find all values of $f(1990)$. $...
0
votes
2answers
30 views

$A$ is a Hermitian operator on an infinite dimensional Hilbert space and $\langle Ax|x\rangle=0$ for all $x$, prove $A=0$ without the spectral theorem

If $A$ is a Hermitian operator on an infinite dimensional Hilbert space such that $\langle Ax|x\rangle=0$ for all vectors $x$, can we prove $A=0$ without the spectral theorem? The proof seems ...
1
vote
0answers
24 views

Is it true that the closed convex hull of a compact subset of the dual equipped with the w*-topology is compact?

Let $X$ be a Banach space. Consider the dual $X^*$ equipped with the weak*-topology. Is it true that the closed convex hull of a compact subset $K$ of the dual $X^*$ is compact? ps: I know that the ...
2
votes
1answer
49 views

Two orthonormal sets in a Hilbert space. One is complete, the other must be complete.

Given two orthonormal sets $\{e_k\}_{k=1,2\ldots}$, $\{e'_k\}_{k=1,2\ldots}$ in a Hilbert space $H$, which satisfy $$ \sum_{k=1}^\infty \|e_k-e'_k\|^2 < 1. \tag{*} $$ Prove that if $\{e_k\}_{k=1,2\...
0
votes
0answers
32 views

Decomposition of spectrum

We know that if X is a banach space and T be an element in Banach Algebra B(X) then the union of residual spectrum continuous spectrum and point spectrum is spectrum of Banach algebra i.e σ(T) is the ...
0
votes
1answer
39 views

boundedness of inverse (Evans PDE)

First, I have difficulty understanding why $$\|u_k\|_{L^2(U)}>k\|f_k\|_{L^2(U)}$$ is being assumed in theorem 6 chapter 6.2 Evans. Second, the last sentence of the proof says (30) implies $\|u\|_{...
0
votes
1answer
28 views

If $A_j$ is an increasing family of Hermitian operators such that $A_j\nearrow A$ weakly, for $A=\mathrm{LUB}A_j$, then $A_j\rightarrow A$ strongly.

I am trying to prove the following proposition from Berberian's 'Notes on Spectral Theory': Proposition 1: If ($A_j$) is an increasingly directed family of Hermitian operators, and if the family ...