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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Exercise 1.19 in Brezis' Functional Analysis

I would like a clarification to an exercise in Brezis' Functional Analysis. Exercise 1.19(2) says Let $E$ be a normed vector space. Let $F:\mathbb R \rightarrow (-\infty, +\infty]$ be a convex ...
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13 views

An exercise about the positive operator

Here is an exercise in functional analysis: An operator $T$ on Hilbert space is positive is positive if and only if all compressions by finite-rank projections ($P_{n}TP_{n}$ for any $n$) are ...
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12 views

supremum of $|\langle\phi,x \rangle_{B^{'},B}|$

Let $(H,\langle \cdot,\cdot \rangle_{H})$ be a separable Hilbert space that is continuously and densely embedded in a separable Banach space $(B,\|\cdot\|_{B})$. i.e. $B'\subset H'=H\subset B$ ...
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17 views

Quotient of a Banach space X gets quotient topology under standard norm induced from X

Let $X$ be a Banach space and $Y$ be a closed subspace of $X$. Then there is a canonical norm on $X/Y$. I want to show that this norm induces quotient topology on $X/Y$. Any hint/solution? I was ...
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18 views

some auxiliary results

I want to prove the following If $N\ge 3$ there exists a constant $c_0=c_0(\Omega)$ such that for all $\alpha\ge 1$ and $z\in H^1(\Omega)$ \begin{align} ...
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On Convex Interpolation and distances

Let $C$ denote the class of all real-valued convex functions on $[0, 1]^2$. Fix $n \geq 2$ and points $x_1, \dots, x_n$ in $[0, 1]^2$. Let $S \subset R^n$ be defined by \begin{equation*} S := ...
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27 views

How to approximate a globally Lipschitz function by differentiable functions with bounded derivatives?

for some positive integer $d \geq 1$ I have a globally Lipschitz continuous function $f \colon \mathbb{R}^d \to \mathbb{R}$ with Lipschitz constant $1$ and would like to approximate it by a sequence ...
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13 views

How to compute the derivative of this functional on a manifold?

I'm a little puzzled by the following computation. Suppose $\Sigma$ is a compact two-dimensional Riemannian manifold and $M$ is any Riemannian manifold. Let $\Omega$ be some space of functions from ...
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6 views

Transitional curves

What are the orthogonal trajectories of a group of curves $f(x,y) + \lambda . g (x,y) = 0$ where $\lambda $ is a connecting real variable, and how to find them?
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15 views

Example of a linear operator pointwise bounded but not uniformly

I was going through the uniform boundedness principle and I need a motivation of a sequence of operator which is Pointwise bounded but not Uniformly. Can anybody help me? Any suggestion is ...
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1answer
45 views

Techniques to solve such a PDE

I have the eigenvalues problem on $[0,\pi] \times [0,2\pi]$ $$\left(\frac{1}{\sin\theta}\frac{\partial}{\partial \theta} \left[\sin\theta \frac{\partial}{\partial \theta}\right] + ...
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18 views

Show that an operator is weakly compact

If $(X,\Omega,\mu)$ is a finite measure space, $k\in L^\infty(X\times X, \Omega\times \Omega,\mu \times \mu)$ , and $K:L^1(\mu)\to L^1(\mu)$ is defined by $$(Kf)(x)=\int k(x,y) f(y) d\mu(y)$$ show ...
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When two projections in a C*-algebra are “almost” Murray-von Neumann equivalent, they are equivalent

Let $A$ be a C*-algebra and $p,q \in A$ be projections. Assume there is an element $a\in A$ such that $\|aa^*-p\|<\frac{1}{4}$ and $\|a^*a-q\|<\frac{1}{4}$. Then there is a partial isometry $v$ ...
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1answer
25 views

Why $f (x):= \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)}$ only belongs to $L^2(0, \infty)$

This is a result given in Royden and Fitzpatrick (p. 143). Show that $$ \int_0^\infty \left[ \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)} \right]^p < \infty $$ if and only if $p=2$. That ...
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6 views

What's the best way to recognize a shape o a function with N-points

I've many shapes with points in theirs countours, how is the best way to recognize a shape? I think the DTF is available but i don't know whether this is the optimal way. P.S. I think if i will ...
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1answer
10 views

Integration over subsets of the complex plane.

Original Problem: Let $\Omega\subset \mathbb{C}$ be an open set and let $f:\Omega\to\mathbb{C}$ be holomorphic such that $f\in L^{2}(\Omega)$. Show that if $B(z,r)$, the ball of radius $r$ ...
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18 views

Fourier Transforms of $L^1$ functions

Suppose that $f_n$ and $f$ are $L^1(\mathbb R^n)$ functions with $f_n \to f$ in $L^1$ sense. Then is it true that their Fourier transforms defined as $$ \hat f(\xi) := \int_{\mathbb R^n} ...
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44 views

Given $M\subset H$ and $\lambda$ a continuous linear functional, show there is a unique linear functional $\Lambda$ on $H$

Suppose $M$ is a closed subspace of a Hilbert space $H$ and $\lambda$ is a continuous linear functional on $M$ with $$\sup_{m\in M, m\neq 0} {|\lambda(m)|\over \|m\|}=c$$ Using Hilbert space ...
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36 views

Projection of a Hilbert space onto orthonormal subset.

Suppose that $\{e_1,e_2,...,e_n\}$ is an orthonormal set in $H$ and define $M\equiv span\{e_1,e_2,...,e_n\}$ Show that $M$ is closed and show that if $P$ is the projection of $H$ onto $M$ then ...
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1answer
12 views

Strong and weak extrema

I am confused about the "strength" of the two definitions. The definitions I use are the following: Let $y$ be a function defined on the set $M$. Neighborhood (0. order) of the function $y$ is the ...
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1answer
22 views

Continuous spectral theorem example

The spectral theorem can be explicitly expressed for an hermitian matrix by providing its eigen decomposition. In the more general case of a bounded self-adjoint operator with a continuous spectrum, ...
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1answer
22 views

Existence of minimum norm solution to linear equation $Tx =y$

Let $T: X \to Y$ be a bounded linear map between Hilbert spaces $(X, \langle \cdot , \cdot \rangle_X)$ and $(Y, \langle \cdot , \cdot \rangle_Y)$ (the Hilbert spaces may be complex or just real ...
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43 views

Is every closed ball (or open ball) in the Eucledean Space $R^n$ convex?

I am solving a problem and I need to use this fact: Every closed ball (or open) in the Eucledean Space $R^n$ convex? Hoever, I am not sure if it is true or not. Can anyone help? Thanks!
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41 views

Composition operator on Sobolev space

I am trying to consider hypothesis on $g$ such that the operator $$ H_0^1 (\Omega) \to H^{-1}(\Omega), \qquad v \mapsto g(v) $$ is $\mathcal C^1$. Of course, $g(0) = 0$. I believe that $g \in ...
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58 views

Exercise 23 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

Consider exercise 23 from chapter 4 ("Hilbert Spaces: An Introduction") of [1] (p. 198). Any help will be much appreciated. Thank you in advance. Suppose $\{T_k\}$ is a collection of bounded ...
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1answer
21 views

Extending mappings on simple tensors

Consider the following situation: Let $H, K$ be Hilbert spaces and let $\Phi$ be some mapping defined on simple tensors in $H\otimes K$ taking values in $B(H\otimes K)$ with the property that each ...
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1answer
39 views

Inner Product in Hilbert Space

Let $H$ be a Hilbert space and $\phi_{1}, \dots, \phi_{n} \in H$ are linearly independent vectors. How can we construct the inner product on $H$ such that $\phi_{1}, \dots, \phi_{n}$ become orthogonal ...
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32 views

Compactness criterion

I have this compactness criterion and I want to apply it, but I don't know what I must write to see if (a) is satisfied and also for (c)? For a subset $H\subset\mathcal{BC}(\mathbb{R},Y)$ to be ...
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32 views

weakly compact operator on $c_0$ is compact

Show that if $T\in {\cal B}(c_0)$ and $T$ is weakly compact, then T is compact. My attempt: $T$ is weakly compact, so there is a reflexive space $X$ , and operators $A\in {\cal B}(X,c_0) $ and $B \in ...
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1answer
18 views

Cardinality of $l_{\infty}$ is c

I need to show that the cardinality of $l_{\infty}$ is $c$, the cardinality of the continuum, where $l_{\infty}$ is the space of all bounded real sequences. Any hint is appreciated. Thanks in ...
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42 views

Continuity of a linear map on Banach space.

A function $f$ on $X$ is said to separate points of $X$ if for $x_1\neq x_2$ then $f(x_1)\neq f(x_2)$. Suppose $A$ is a subset of $Y^{*}$, the space of functionals on a Banach space $Y$, which ...
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21 views

Proving a set is a vector space, proving a norm, and that the set with the norm is a Banach space

Let $c$ be the space of sequences of real numbers that converge. That is $x\in c$ means that $x=(x_1,x_2,...)$ and $lim_{j\to \infty} x_j$ exists. It is easy to verify that $c$ is a vector space. For ...
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29 views

If any two norms on a vector space are equivalent then the space is finite-dimensional [duplicate]

I need to prove: If any two norms on a vector space are equivalent then the space is finite-dimensional. I am aware of the converse of this result that on a finite dimensional vector space any two ...
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1answer
35 views

Appoximation of Lipschitz functions by $C^1-$functions

I came across the following statements in a math book without proof. Denote $M_k$ as the set of functions from $C[a,b]$ that is K-Lipschitz continous. i.e $\forall x,y,|f(x)-f(y)|\le K|x-y|$ 1) The ...
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32 views

What does this symbol mean?

I am studying a book on functional analysis and came to a definition that started like this: Let $M$ be a set and $F: M \hookleftarrow$ a function (...) My only question is: what does the ...
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1answer
41 views

Dimension for a closed subspace of $C[0,1]$.

Let $X \subset C^1[0,1]$ be a closed subspace of $C[0,1]$ (with sup norm). Prove that $X$ has to be finite-dimensional.
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1answer
28 views

Relationship between Characteristic Function and Eigenfunction

In probability we talk about "characteristic functions" of random variables, usually written as $\Phi_X(t)=E[e^{itX}]$. Is the characteristic function in some sense an "eigenfunction" (a function f ...
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34 views

At most n functions

Some background: I was trying to solve the functional equation f(f(x))=sin(x). I realized that $f(\pi n)$ is a root of f for all integers n, because $f(f(\pi n))=\sin(\pi n)=0$. Thus, we can write f ...
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1answer
18 views

The continous of a function in the Sobolev class

Let $f\in S$ with $S= \left\{ {f:\mathbb{R} \to \left[ {0, + \infty } \right):\int_{ - \infty }^{+\infty} {{{\left| {\hat f\left( t \right)} \right|}^2}{{\left( {1 + {{\left| t \right|}^2}} ...
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1answer
44 views

Closure of $C_0^{\infty}$ in $W^{k,p}(\Omega)$

Why is it that in the definition of $W_0^{k,p}(\Omega)$ for $\Omega$ with boundary smooth enough, we only have $D^{\alpha}u$ for all $0\leq|\alpha|\leq k-1$ vanishing at the boundary and not ...
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Dense invariant domain stable under resolvent?

I have thought about the following problem: Let $A_1\dots A_n$ a family of (unbounded) essentially selfadjoint operators on some Hilbert space $\mathcal{H}$ and $\Phi\subset\mathcal{H}$ the maximal ...
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Norm of a linear function on $c_{0}$

Let $c_0$ be the space of all sequences which converge to $0$. Let $f: c_0 \to\mathbb{R}$ by $$f(x) = \sum_{n=1}^\infty \frac{x_n}{n^2}$$ What is the value of $\|f\|$?. I am just getting used to the ...
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1answer
27 views

What sequences could satisfy these requirements?

I need to find a sequence which converges to $0$ but is not in any space $\ell^p$, where $1 \leq p < +\infty$. And, I need to find a sequence which is in every space $\ell^p$ with $p > 1$ but ...
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3answers
51 views

Basis of eigenfunctions in Banach space

I have a question about proving the existence of a basis of eigenfunctions. Assume we have a compact operator $L:\mathcal{B}\rightarrow\mathcal{B}$, where $\mathcal{B}$ is a Banach space of analytic ...
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1answer
56 views

Exercise 31 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

The following problem shows that $\{e^{inx}\}_{n \in \mathbb{Z}}$ is an orthonormal basis of $L^2([-\pi, \pi])$. It is taken from [1] (exercise 31 of chapter 4: "Hilbert Spaces: An Introduction", pp. ...
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45 views

Prove that $C[a,b]$ with inner product $\langle f,g\rangle:=\int_a^bf(t)\overline{g(t)}dt$ is not a Hilbert space.

Prove that $C[a,b]$ with inner product $<f,g>:=\int_a^bf(t)\overline{g(t)}dt$ is not a Hilbert space. Now the norm induced by the inner product is \begin{align} ...
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1answer
17 views

Proving that the 1-norm, $||x||_1$ is not generated by inner products on $\mathbb{C}^n$

Proving that the 1-norm, $||x||_1$ is not generated by inner products on $\mathbb{C}^n$. Is it sufficient to take $x=(1,0)$, $y=(0,1)$ in $\mathbb{C}^2$ and just showing that \begin{align} ...
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1answer
33 views

Are there any interesting Hilbert spaces that do not present as function spaces?

I was pondering this question in class earlier: All separable, infinite dimensional Hilbert spaces are isometrically isomorphic. Thus, in particular, any such space is isometrically isomorphic to ...
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20 views

How Many Negative Eigenvalues of $-\frac{d^{2}}{dx^{2}}$ on $[0,L]$?

What is the maximum number of eigenvalues $\lambda < 0$ for the trigonometric problems?: $$ \begin{array}{c} -\frac{d^{2}f}{dx^{2}}=\lambda f,\\ ...
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1answer
29 views

Ask for a good reference for the calculus involving singular continuous measure

I am not an expert on measure theory. I am sorry if this question is too simple for some experts here. Suppose the measure $\mu$ is singular continuous on $\mathbb{R}$, such as the cantor measure. ...