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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22 views

Prove for the family of operators $\{S(t)\}_{t>0}$ that $S'(0)$ exists

Let's consider a real-valued function $V\in L^\infty(\mathbb{R})$ and a family of operators $\{S(t)\}_{t>0}$ defined on $L^2(\mathbb{R})$ as follows: $$\qquad \qquad\left(S(t)f \right)(x) = \frac{...
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1answer
21 views

A question on uniform algebras

Let $A$ be a uniform algebra on a compact metric space $X$ Why the necessary condition for $A$ to be $C(X)$(the algebra of all complex-valued continuous functions on $X$) is that the maximal ideal ...
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1answer
29 views

Question on Inequality from Bartle's Elements of Integration: Riesz Fischer Theorem

I am puzzled how did Bartle get $$|g_k|\leq\sum_{j=k}^\infty |g_{j+1}-g_j|$$ (second last line)? I tried using Triangle Inequality and ended up with one extra term: $$\begin{align*} |g_k|&=|g_k-...
4
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1answer
46 views

the proof of variational principal for the principal eigenvalue (checking orthonormal subset)

Hi I am looking at part 3 of the proof in Evans Chapter 6. I have difficulty understanding "Furthermore from (6) and (7) we see that $(\lambda_k^{-1/2} w_k)$ is an orthonormal subset of $H_0^1(U)$. ...
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33 views

Sobolev Space with partial inner product

In my work, I encountered the following problem. Consider the set of real-valued functions, which are ``balanced'', that is the set of bounded functions $f(x)$ such that $\lim_{x\rightarrow \pm \...
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2answers
52 views

$||T-I|| < 1$ implies that $T$ is invertible.

Let $B$ be a banach space and $T : B \to B$ be a bounded linear transformation. If for identity transformation $I : B \to B$ , $||T-I||$<1 , then $T$ is invertible. || || is norm of ...
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43 views

Show that usual $L^2$ norm is equivalent to arbitrary norm $||\cdot||$ that satisfies 'convergence condition'

Given $L^2(\mathbb{R})$ consider a norm $||\cdot||$ on $L^2$ such that $(L^2,||\cdot||)$ is a Banach space and every $||\cdot||$-convergent sequence has a subsequence that converges almost everywhere. ...
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1answer
12 views

Is the delay line and bounded operator in a normed space?

Let $X$ be the normed space of all bounded real-valued functions on $\mathbb{R}$ with norm defined by $$ \|x\| = \sup_{t \in \mathbb{R}}{|x(t)|} $$ and let $T: X \rightarrow X$ be defined by $$ y(t)...
4
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2answers
84 views

Supremum of absolute value of the Fourier transform equals $1$, and it is attained exactly at $0$

Suppose that $f \in L^1(\mathbb{R}^n)$, $f \ge 0$, $\|f\|_{L^1} = 1$. How do I see that $\sup_{\xi\in\mathbb{R}^n} |\mathcal{F}(f)(\xi)| = 1$, and it is attained exactly at $0$?
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1answer
38 views

An exemple of strict inequality for reverse inequality Minkowski for space $L^p$, $0 < p <1$

Let be $0<p<1$. Suppose that we know that $$ \bigg(\int (u + v)^p\bigg)^{1/p} \geq \bigg(\int (u)^p\bigg)^{1/p} +\bigg(\int (v)^p\bigg)^{1/p}$$ for all $u,v \geq 0$ in $L^p$. I need find an ...
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2answers
23 views

Representation of functional on overlapping areas

I have given a functional $l$ on $C_c^\infty(\mathbb{R}^n)$. Now let's assume that for any $p \in \mathbb{R}^n$ we have a neighborhood $V_p$ and a $2\pi$-periodic $C^\infty$-function $u_p$ on $\mathbb{...
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1answer
45 views

Proving $\{T(x_n) \}$ converges to $0$.

Let $X$ and $Y$ be normed spaces and $T: X \rightarrow Y$ a linear operator. Prove that if the graph of $T$ is closed then $\{Tx_n \}$ converges to $0$ in $Y$, for all sequences $\{x_n \}$ that ...
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0answers
89 views

Continuous embeddings

Given the following exercise: We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \overline{\{x \in (\mathbb R_+) : f(x) \neq 0\}} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$ (the ...
3
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1answer
31 views

About equivalent norms on a vector space

Definition. A norm $\|\cdot\|$ in a vector space $X$ is said to be equivalent to a norm $\|\cdot\|_0$ on $X$ if there are positive numbers $a$ and $b$ such that for all $x \in X$ we have $$ a\| x \|...
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1answer
44 views

If $\int_0^1 |f|dx= (\int_0^1|f|^p dx)^{1/p}$ for some $p > 1$, then $f$ is constant.

If $\int_0^1 |f|dx= (\int_0^1|f|^p dx)^{1/p}$ for some $p > 1$, then $f$ is constant. Since $\int _0 ^1 |f| dx = \int _\mathbb{R} fg dx $ where $g \mbox{ is characteristic function on } [0,1]$, I ...
4
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1answer
33 views

proving an element is unitary in a C* algebra

Let $p,q$ be projections in a unital C*-algebra $A$ and let $\tilde{A}$ be the unitization. I'd like to show that if $p\sim_u q$ (ie $q=zpz^*$ for $z$ unitary in $\tilde A$), then $q=upu^*$ for $u$ ...
7
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1answer
64 views

Prove that there exists a sequence $(x_n)$ such that $\sum_n a_n x_n$ diverges

So, here's a nice little result that I deduced using the closed graph theorem from functional analysis, but I'm wondering if there's a more elementary approach: Fact: Let $(a_n)$ be a sequence ...
3
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1answer
24 views

Proving linearity of an operator using boundedness.

I am considering an operator $K\colon \ell^2 \to \ell^2$ given by $$Kx = \sum_{n=1}^\infty e^{-n} \langle x , e_n\rangle e_n $$ where $e_n = (\delta_{k,n})_{k\in \mathrm{N}}$ is the standard basis on ...
4
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1answer
30 views

$\mathcal{l}^1$ is not complete for the norm $\|\cdot\|_\infty$

Let $\mathcal{l}^\infty = \{ (u_n) | u_n \in \mathbb{R}$ and $sup_{n \in \mathbb{N}}|u_n| < \infty \}$ and $\mathcal{l}^1 = \{ (u_n) | u_n \in \mathbb{R}$ and $\sum_{n=1}^{\infty} |u_n| < \infty ...
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0answers
58 views

A basis for $\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\}$, and how compute coordinates

Let $$\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\},$$ where we take $$\eta_\alpha(t)= \left\{ \frac{\alpha}{t} \right\} -\alpha \left\{ \frac{1}{t} \right\},$$ and $ \left\{ x ...
4
votes
2answers
61 views

Does existence of the second weak derivative of $f\in L^2$ imply existence of the first?

Let's consider a function $f\in L^2(\mathbb{R})$ for which the second weak derivative exists and lie in $L^2(\mathbb{R})$, i.e. there exists $f''\in L^2(\mathbb{R})$ such that for all $\varphi\in C_0^\...
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0answers
14 views

A comparison between absolute values of functionals

Let $A_0$ be a C*-subalgebra in a C*-algebra $A$. Let $\phi_0$ be a bounded linear functional on $A_0$ and assume $\phi$ is an extension of $\phi_0$ on $A$. I mean $\phi\in A^*$ with $\phi_{|_{A_0}}=\...
0
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1answer
32 views

Good, simple reference for Riesz-Fischer Theorem.

I am looking for a good, simple reference for the proof of Riesz-Fischer Theorem ($L^p$ spaces are complete). An example of a not so good reference in my opinion is Royden, where he uses "rapidly ...
6
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1answer
47 views

Norms on an Ultraproduct

Suppose $X$ is a Banach space and $\mathcal{U}$ is a non-principal ultrafilter on $\mathbb{N}$. I am interested in the Banach space $(X)_\mathcal{U}$, where we consider sequences $(x_i)_{i \in \mathbb{...
1
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1answer
36 views

variational principle for the principal eigenvalue

I am reading the proof of theorem 2 in chapter 6 Evan PDE. I have difficulty verifying the following part of the proof, i.e. 3 questions here. 1) The assumptions $u\in H_0^1(U)$ and $u\in L^2(U)=1$ ...
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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
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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.
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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)\...
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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
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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? $$\|\vec{u}\|_{\infty}=\sum_{i=1}^3\sup|u_i|.$$
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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
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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
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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 ...
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157 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{...
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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^*...
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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 $X,Y$ ...
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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
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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
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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 ...
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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
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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
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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"),...
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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
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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
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1answer
27 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) = ...