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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Ask for reference convergence of implicit euler method for initial value problem with dissipative source term

I am considering the convergence of implicit euler method for solving the following initial value problem: \begin{cases} u'(t)=f(t,u(t)),t\in[0,T]\\ u(0)=u_0\in \mathbb{R}, \end{cases} where ...
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2answers
10 views

Example to show that unconditional convergence does not imply absolute convergence in infinite-dimensional normed spaces.

I tried to show that unconditional convergence does not imply absolute convergence in infinite-dimensional normed spaces using a direct proof, but unfortunately I did not succeed. The definition of ...
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1answer
19 views

Space of bounded linear operators might fail to be a Banach space.

I tried to show that the space of bounded linear operators $B(X,Y)$, where $X$ and $Y$ are normed linear spaces, might fail to be a Banach space. To show this, I considered the space $X = \ell^1 ...
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11 views

Compact set of function involving BSpline functions.

Let $X = \left\{x_0,...,x_{n-1}\right\}$, $x_i-x_{i-1} = h$ for $i=1\ldots n-1$ and $$ \phi(x;X,\vec{\beta}) = \sum_{j=0}^n \beta_j \phi_{j,2}(x;X) $$ where $\phi_{j,2}$ is the $j$-th BSpline (of ...
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16 views

global inverse function theorem

Consider a continously differentiable non-constant function $f:\mathbb{R}^2\to\mathbb{R}$. Define $$ K=\{x\in\mathbb{R}^2|f(x)=0\}. $$ I wish to know whether there is a continuously differentiable ...
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16 views

Hausdorff-Quotient: Embedding

Problem Given a uniform space $\Omega$. (Exemplary Topological Vector Space!) Consider a dense subspace: $$\iota:\mathcal{D}\hookrightarrow\Omega:\quad\overline{\iota\mathcal{D}}=\Omega$$ Regard ...
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1answer
20 views

If $T(B)$ is a bounded subset of $K$, then, is the linear functional $T$ bounded (norm of $T$ is finite)?

Let $H$ be a infinite dimensional Hilbert space on $K$ and let $B$ be a basis of $H$ that $H=\overline{span(B)}$ moreover, let $T : H \rightarrow K$ be a linear functional (i don't know if it is ...
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35 views

Abount linear functional: If $T(B)$ is bounded, is $T$ bounded?

Let $H$ be a infinite dimensional Hilbert space and let $B$ be a basis of $H$ that $H=\overline{span(B)}$ moreover, let $T : H \rightarrow K$ be a linear functional If $T(B)$ is bounded, is $T$ ...
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3answers
19 views

Define injective function and prove by giving an example.

Answer: A function is injective if and only if whenever $f(x)=f(y)$, $x=y$; for all $x,y \in \text{dom}(f)$ my question can someone provide an example proving the above definition that $f(x)=f(y),\ ...
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1answer
35 views

Which sets are dense in $C([0, 1])$? [on hold]

Which of the following sets are dense in $(C[0,1], ||\cdot||_\infty)$ (the space of real valued continuous functions on [0,1] with respect to sup-norm topology)? $\{f \in C[0,1]\ |\ f \text{ is ...
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8 views

Is the derivative product rule true for Bochner spaces?

If $u\in L^{2}(0,T; L^{2}(\Sigma))$ with $u_{t}\in L^{2}(0,T; H^{-1}(\Sigma))$ and $v\in L^{2}(0,T; H^{1}_{0}(\Sigma))$ with $v_{t}\in L^{2}(0,T; L^{2}(\Sigma))$ is it true that $$ ...
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0answers
18 views

Name or reference about a inequality with integrals?

I have wrote down some class notes and I think I copied something wrong. It is an integral inequality; $$\iiint_{B^n}|\nabla\psi|^2\frac{1}{|x|^{n-2}}dV\leq C\iint_{\partial B^n}|\psi|^2dA$$ where ...
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1answer
24 views

Solution of a sublinear elliptic problem.

In a lecture notes, the author showed the problem $\tag{$P$}$ $\begin{cases} -\Delta u = |u|^{q-2}u \textrm{ in } \Omega, \\ u(x)= 0 \textrm{ in } \partial\Omega, \end{cases}$ where $\Omega ...
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1answer
23 views

Prove that a set which satisfies every nonempty subset contains a weak Cauchy sequence must be bounded

I got stuck on this problem about weak convergence in normed space. This problem is exercise 9, page 263 in Functional Analysis of Erwin Kreyszig. So the question is basically on the title, let me ...
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2answers
31 views

$K$ compact metric space, is there a finite set of continuous functions that separates points in $K$?

Definition: A family of functions $\mathcal{F}$ on a set $X$ separates points in $X$ if for every distinct pair $x,y\in X$ there exists $f\in\mathcal{F}$ such that $f(x)\neq f(y)$. Let $K$ be a ...
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1answer
37 views

Why does $a(1_A-x)=(1_A-x)a=1_A$

I am working through the following theorem and proof, but I struggle to understand the last part. Theorem: Let $A$ be a Banach algebra with unit element $e$, $x \in A$ and $||x||< 1$. Then ...
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3answers
27 views

Ordering: Supremum

Given a C*-algebra $\mathcal{A}$. (It may or maynot be unital!) Increasing Sequence: $$A_n\leq A_{n+1}\leq\ldots\leq L:\quad\lim_nA_n\in\mathcal{A}$$ Ordered Family: $$A_n\leq A_{n+1}\leq\ldots\leq ...
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24 views

About isometry on Banach spaces

Let $X$ be a Banach space. For every $x\in X,$ the non-empty dual duality set $\mathcal{J}(x)$ is defined as:$$\mathcal{J}(x):= \left\{j(x) \in X': \langle x, j(x)\rangle = \|x\|^{2} = \|j(x)\|^{2} ...
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1answer
18 views

Intersection of normed speces and desity

Let $(X_n, \|\cdot\|_n)$ be a sequence of normed spaces. My first question is, whether it is possible to norm $X= \cap_n X_n$. My idea would be to take $\|\cdot\|_X = \sup \|\cdot\|_n$ if it is ...
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1answer
35 views

Why is the following scaling good for the general case?

In the following post how come scaling to $x_0=0$ and $r=1$ can help to prove for any $r,x_0$? Is there a general rule for when is it OK to scale?
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1answer
37 views

Question on proof of unit ball in $C([0, 1])$ not being compact

Take the sequence $f_n(t)=t^n$, $0\le t\le 1$. Then $\{f_n\} \subset \overline{B(0,1)}$, but we have no subsequence of $\{f_n\}$ converging in $C([0,1])$. So the unit ball is not compact in ...
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12 views

Definition of $L^p(C,B)$ where $C$ is a subset of the domain of the function and $B$ is a Banach space.

I am looking for the definition of $L^p(C,B)$ as declared in the title. I know that $C^s(C,B)$ is defined with the norm: $$\frac{ \| f(x) - f(y)\|_B}{|x-y|^s}\le A$$ i.e $f\in C^s(C,B)$ iff the ...
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1answer
21 views

How to show $\sigma(T_q) = \overline{\{q(t) : t \in [0,1]\}}$ where $T_q$ is the multiplication operator?

Let $B$ be the Banach space of bounded complex functions on $[0,1]$ with sup-norm. For $q \in B$, define the (multiplication) operator $T_q : B\rightarrow B$ by $(M_q f)(t) = q(t)f(t)$. How do you ...
3
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1answer
60 views

Show that $\|f_1+f_2\| \leq \|f_1\| + \|f_2\|$ using Minkowski's inequality

I am trying to show that: $\|f_1+f_2\| \leq \|f_1\| + \|f_2\|$ using the Minkowski inequality. for: $$ \|f\| = \left(\int_0^1 \left[|f|^2 + |f'|^2\right]\ dx\right)^{1/2}.$$ I dont see how I can ...
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1answer
67 views

In Rudin's proof of the completeness of $L^\infty$

This is closely related to a previous question: In a proof of the completeness of $L^\infty$ The following is a proof of completeness of $L^\infty$ by Rudin in his Real and Complex Analysis: Here ...
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86 views

In a proof of the completeness of $L^\infty$

The following is a proof of the completeness of $L^\infty$ in a lecture note by Hunter: Here are my questions: Can one (literally) replace $1/m$ in the whole proof with $\epsilon$ and replace ...
2
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0answers
24 views

How do linear operators acting on paths of Gaussian processes influence the covariance function?

It is well-known that applying a linear transformation $A$ on an $n$-dimensional centered Gaussian distribution with covariance matrix $\Sigma$ results in another centered Gaussian distribution with ...
2
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1answer
55 views

How can I prove $\lim_{\epsilon \to 0} \space \text{Im}\frac{1}{x+i \epsilon}=-\pi\delta(x)$?

I want to show that: $$\lim_{\epsilon \to 0} \space \text{Im}\frac{1}{x+i \epsilon}=-\pi\delta(x)$$ This is my attempt: I assumed that $\text{Im}$ stands for the imaginary part. Therefore, ...
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0answers
27 views

If $p{\le}r$, then r-norm of x is smaller then p-norm of x, where x belongs to lp..

It is evident that $l^p$ is a subset of $l^r$. $l^p$ is space of all sequences, such that $p$-sums are convergent. It is a special case of $L^p$ with $X$ as $\mathbb{N}$ and numbering measure. How ...
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22 views

Structured singular value bound on norm of an inverse operator

My question is related to the first answer here, Norm of an inverse operator: $\|T^{-1}\|=\|T\|^{-1}$? with the difference being that I want a similar bound (if possible) with the structured ...
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1answer
22 views

$L^2$ - base functions

say we have a function, $f$, in $L^2$ and have different base functions for $L^2$, then is there a reason to believe that one base is "better" compared to the other when writing expansions for $f$? ...
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1answer
36 views

How does the compactness property help us show a subset $A$ of a metric space $X$ is closed?

We have a compact subset $A$ of a metric space $X$ and we want to show that this implies that $A$ is closed. Let $y \in A$ and $y \in A^c$. For each $y \in A$, we can take open neighbourhoods $U_y$ ...
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2answers
34 views

What is Convex about Locally Convex Spaces?

This might be a silly question, but what motivates the name "locally convex" for locally convex spaces? The definition in terms of semi-norms seems to have nothing to do with convexity or with the ...
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35 views

Can I have a map to follow in this proof?

Let X be a normed vector space, $Y:=B(X,\mathbb{K})$ a vector space over $\mathbb{K}$ of the bounded linear functionals with the usual norm $$||f||=sup_{||x||=1, x\in X}|f(x)|, f\in Y$$ and$$F(x) ...
2
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1answer
23 views

Convexity of Hilbert cube [on hold]

I am trying to show that the Hilbert cube $\{ x_n \in l^2(\mathbb{N}) \mid x_n \in [0, \frac{1}{n}] \ \forall n \in \mathbb{N} \}$ is convex and (norm)-compact.
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1answer
49 views

Cardinality of a Hilbert space

I have seen the theorem about the cardinality of orthonormal basis of a Hilbert space. I wonder if we have a Hilbert space $H$ with an orthonormal basis having cardinality of the continuum, then what ...
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1answer
49 views

Bounded operator on Hilbert space

Let $H$ is a Hilbert space. If $T\in B(H)$ show that $T+T^*\ge 0$ iff $T+I$ is invertible in $B(H)$ with $\|(T-I)(T+I)^{-1}\|\le 1$. (Hint is $T+T^*\ge 0$ iff $\|(T+I)x\|\ge \|x\|\ $ and ...
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1answer
41 views

How much does the $L^p$ norms say about a function?

Let's say we have two positive, decreasing function $u$ and $v$ on $[0,+\infty)$, and we know that $\|u\|_{L^p}=\|v\|_{L^p}$ for all $p\ge1$, can we say something about $u$ and $v$? Do they have to be ...
2
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2answers
38 views

If for $u \in L^2(\mathbb{R}^n)$ , we define $v(t)=u(x+th) $ $v: [0,1] \to \mathbb{R}$ $\Rightarrow^?$ $v \in L^2((0,1))$

I have a function $u(x) \in L^2(\mathbb{R}^n)$ ($n \geq 2$). Suppose we define another function $v$ as $$v:[0,1] \to \mathbb{R} $$ $$\quad \quad \quad \quad \quad \quad \ t \to u(x+th)$$ where $h \in ...
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43 views

what is the ODE that gives rise to the Laplace Transform.

It can be shown that the transform pairs can be obtained from a differential equation with some boundary conditions. See, for example, Keener's book chapter 7. For example. The Fourier transform can ...
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1answer
19 views

Closed Graph Theorem on Finite Dimensional Banach Spaces

My professor wants us to prove that every linear mapping from a finite dimensional Banach space is continuous. BUT, he wants us to do so using the Closed Graph Theorem (in functional analysis). ...
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31 views

Suppose : $E:R \longrightarrow R$ [on hold]

Suppose : $E:R \longrightarrow R$ if $E$ : $$E(a+b)=E(a)E(b): \forall a,b\in R$$ then prove : $$E(r)= E(1)^{r}; \forall r\in Q $$ $Q$= Rational numbers thank you !
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1answer
49 views

Prove a subspace of $\mathbb R^n$ is closed

I want to prove $C=\{v=(x,y)|x,y\geq 0\}$is a closed subspace of $\mathbb R^2$. Basically I want to generalize the suggestion in this post for any $\mathbb R^n$: Do I need to show that $v_n ...
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2answers
60 views

Is the Hardy space $H^\infty$ closed under differentiation? [duplicate]

** As it's been pointed out, the underlying matter of this question (that $f$ holomorphic and bounded does not imply $f'$ bounded) has been already answered in other post. However, here it's seen as a ...
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1answer
34 views

How to evaluate the Lebesgue integral of the Heaviside function?

I have to evaluate the Lebesgue integral $$ I = \int\limits_{[-1, 1]} \chi(x) \chi(x - \frac{1}{2}) d\left(\chi(x)\chi(x + \frac{1}{2})\right) $$ where $ \chi $ is the Heaviside function: $$ ...
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1answer
20 views

Closest element to a subset of $\mathbb R^2$

Let $U=\{(x,y)|x,y\geq 0\}$ be a closed convex subspace of $(\mathbb R^2,\|\cdot\|_\infty )$. Show that the closest elements in $U$ to $(1,-1)$ are $\{(x,0)|0\leq x\leq 2\}$ Show that the closest ...
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1answer
16 views

A bounded linear functional on a Hilbert space that is a Hahn-Banach extension of one on a subspace

Let $M$ be a closed linear subspace of a Hilbert space $H$ and $g\in M*$(all bounded linear functional on $M$). Let $\pi$ be the orthogonal projection of H onto M, then $f=g\circ\pi$ is the only ...
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32 views

Convergence on $\ell^p$ spaces

I have been working on this problem. Show the following are equivalent: i. $\sup_{n\in\mathbb{N}}\|x_n\|_p<\infty$ and $\lim_{n\to\infty}x_n(j)=0$ for each $j\geq1$, ii. ...
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17 views

Can dimension counting argument generalize to functional space

By dimension counting I mean the following argument: there is no injective continous mapping from any open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$ if $n>m$ (is this true? I can't give rigous ...
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2answers
74 views

Prove for a close subset of $\mathbb R$

Let $S=\{x\in \mathbb{R}\mid 0\leq x\}$. Prove that $S$ is a closed subset of $\mathbb{R}$. I know I need to show that $\forall x \exists x_n \xrightarrow[n \to \infty]{} x\in S$, but I have no ...