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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Proving that a subspace of a normed vector space is closed

Question: Let X be a normed vector space. If M is a closed subspace of X and x ∈ X − M then M + ℂx is closed where M + ℂx = { y + λ x : y ∈ M , λ ∈ ℂ } There's a theorem from Folland's Real Analysis ...
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16 views

How to determine a function from its corresponding distribution?

If we have a function $\phi(x)$ we can determine the corresponding distribution $\phi^D$ such that: $$\forall f:L_{\phi^D}(f)=\langle\phi^D|f\rangle=\int_\mathbb{R}\phi(x) f(x) dx$$ as long as $f$ ...
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37 views

Show that $g\in L^1(\mathbb{R})$ but $g^2\not\in L^1(\mathbb{R})$.

$$g(x) = \begin{cases} 1/\sqrt{x} & \text{ for }0<|x|<1,\\0 & \text{ otherwise.} \end{cases}$$ Show that $g\in L^1(\mathbb{R})$ but $g^2\not\in L^1(\mathbb{R})$. Wouldn't $g^2$ just be ...
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26 views

A problem on real analysis in which from derivative you need to tell about function.

Let $f:(0,\infty)\to\mathbb{R}$ be differentiable.if $f'(x)\to l $ as $x\to\infty,$then show that $\frac{f(x)}{x}\to l $ as $x\to\infty$ . i have no idea where to start.any hint please
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25 views

Are exponential functions a complete system on $L^2\left(\mathbb{T},\mu\right)$

Let $\mathbb{T}$ denote the unit circle in $\mathbb{C}$, and let $\mu$ be a complex Borel measure on $\mathbb{T}$. Consider the space of square integrable functions $L^2\left(\mathbb{T},\mu\right)$. ...
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46 views

Bounded sequence in a normed space converges weakly

Can anyone help me here? Question: "X is a normed space and A is a subset dense in the dual of X. x belongs to X and the sequence (x_n) of X is bounded of E such that f(x_n) converges to f(x) for all ...
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461 views

Dual Bases: Finite VS Infinite Dimensional Spaces

Motivation I know that in finite dimensional spaces like a $n$ dimensional Euclidean space $\mathbb{E}^n$, for every basis $G=\{g_1,g_2,...,g_n\}$ we can define a dual basis $G'=\{g^1,g^2,...,g^n\}$ ...
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36 views

Show that if $f$ is a uniformly continuous function on $\mathbb{R}$ and $f\in L^1(\mathbb{R})$, then $f$ is bounded and $\lim_{|x|\to\infty}f(x)=0$.

Show that if $f$ is a uniformly continuous function on $\mathbb{R}$ and $f\in L^1(\mathbb{R})$, then $f$ is bounded and $\lim_{|x|\to\infty}f(x)=0$. I'm not entirely sure what I should be doing. ...
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67 views

Prove that $l^2$ is closed and bounded but not compact.

Consider the space $l^p=\{(x_i);x_i\in \mathbb C:\sum |x_i|^2<\infty\}$ .Define a norm on $l^2$ by $||x||=\sqrt{\sum |x_i|^2}$. Prove that $l^2$ is closed and bounded but not compact. I know ...
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34 views

If $A$ is a positive operator and $B$ is a bounded operator, show that $B^*AB$ is positive.

If $A$ is a positive operator and $B$ is a bounded operator, show that $B^*AB$ is positive. Where both $A$ and $B$ are operators in a Hilbert space $H$. I know that if $A$ is a positive operator, ...
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111 views

Why is this set compact in $L^2(\mathbb{N})$?

Suppose $L^{2}(\mathbb{N})$ is the Hilbert space of sequences $(a_{n})_{n \in \mathbb N}$ which satisfy $\sum |a_{n}|^{2}$ with $(a,b) = \sum a_{n} \bar{b_{n}}.$ Prove the set of sequences $\{a_{n}\}...
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61 views

Relationship among the function spaces $C_c^\infty(\Omega)$, $C_c^\infty(\overline{\Omega})$ and $C_c^\infty(\Bbb{R}^d)$

I have seen the spaces $C_c^\infty(\Omega)$, $C_c^\infty(\overline{\Omega})$ and $C_c^\infty(\Bbb{R}^d)$ a lot in theorems regarding PDE where $\Omega$ denotes some open subset of $\Bbb{R}^d$. There ...
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44 views

How is $\lim_{\rho \to 0} \rho^{1 - n} \int_{\partial B_\rho} u ds = n \omega_n u(y)$?

I'm looking at some material on harmonic functions and it says $$\lim_{\rho \to 0} \rho^{1 - n} \int_{\partial B_\rho} u ds = n \omega_n u(y)$$ where $n$ is the number of dimensions and $\omega_n$ ...
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22 views

semi-continuity of a function on $H_0^1(\Omega) $ with respect to the weak topology

In the sobolev space $H_0^1(\Omega)$, prove that the function $$ h(v)=\frac12||\nabla v||_2^2-\int_\Omega f(x)v(x)dx $$ is semi-countineous with respect to the weak topology, where $f\in L^2(\Omega)$,...
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61 views

Does $\|\sum_{i=1}^{\infty}x_i\| \leq \sum_{i=1}^{\infty}\|x_i\|$ hold for a norm?

In a normed linear space $(X,\|\cdot\|)$, by definition we have $\forall x,y \in X$, $$\|x+y\| \leq \|x\|+\|y\|.$$ My question is, is it true (or does the definition of norm imply) that for a sequence ...
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35 views

Taking $\inf$ for sobolev space in different order

Let $\Omega\subset \mathbb R^N$ open bounded, smooth boundary be given. Define $$ F(u,v):=\int_\Omega |\nabla u|^2v^2dx+\int_\Omega(|\nabla v|^2+(1-v)^2)dx, $$ and two sets $\mathcal U:=\{u\in H^1(\...
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1answer
47 views

Characterize the continuity of linear maps between Banach spaces in terms of continuous linear functionals

I solved a) and b). But I cant seem to get a grip of what characterizes the functionals for which we want continuity in the general case. Hints please!
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1answer
29 views

Normed Quotient Space

If $X$ is a normed vector space and $M$ is a proper closed subspace, I want to show that for any $\epsilon>0$ there exists an $x\in X$ such that $\|x\|=1$ and $\|x+M\|\geq 1-\epsilon$. Is there ...
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51 views

Closure of a differential operator

Consider $A:\mathcal{D}(A)\subset L^{2}[0,1]\to L^{2}[0,1]$ given by $$A:=-\frac{d^{2}}{dx^{2}},\qquad\mathcal{D}(A):=C^{2}_{0}(0,1)$$ Now, I assume that the closure of $A$ is its extension defined ...
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The sequence of functions $f_n=\frac{n}{1+n\sqrt{x}}$ in $L^1(0,1)$ and $L^2(0,1)$

During a lecture on $L^p$ spaces, the lecturer made a few comments about the sequence of functions $f_n=\frac{n}{1+n\sqrt{x}}$ that I am not sure I fully understand. 1) First he said $f_n\in L^2(0,1)...
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2answers
35 views

What is the difference between n-dim ($n<\infty$) Banach space and $R^n$?

What is the difference between n-dim ($n<\infty$) Banach space and $R^n$? I feel they are same ,no matter under homeomorphism isomorphism or diffeomorphism. In fact, I feel they are not ...
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1answer
24 views

A bounded family of functions in $L^p[E]$, where E is a measurable set, is uniformly integrable.

A corollary in Royden & Fitzpatrick's Real Analysis (chapter 7 section 2) reads: Let $E$ a measurable set, and $1<p<\infty$. Suppose $F$ is a family of functions in $L^p(E)$ that is bounded ...
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2answers
47 views

Show $\|A\|=\sup_{x\neq 0} \langle Ax,x\rangle/\langle x,x\rangle$ for a positive operator $A$.

I have a positive operator $A$ on the Hilbert space $\mathcal{H}$. I must prove that $\|A\|=\sup_{x \ne 0}\frac{(Ax,x)}{(x,x)}$. I am only able to get one inequality: Assume $x$ is nonzero: $$\frac{...
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Quasiconvex and lower semicontinuous function

Definition 1. Let $X$ be a Banach space and $f:X\rightarrow\overline{\mathbb{R}}$. The function $f$ is said to be proper if $f(x)>-\infty$ for all $x\in X$ and there exists $x_0\in X$ such that $f(...
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1answer
46 views

the relationship between convergence and topology

I found the following in a textbook: The topology in $C^\infty(\mathbb R^n)$ is defined by: If $\{\varphi_\nu\}$ satisfies that for all compact sets $K$ and for all multi-indices $\alpha$, $\sup_{x\...
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42 views

Show that if $f$ is a bounded function on $E$ and $f\in L^{p_1}$ then $f\in L^{p_2}$ for any $p_2>p_1$. [duplicate]

I'm working through Royden & Fitzpatrick's Real Analysis, and one of the questions in the introductory chapters of $L^p$ spaces reads: Show that if $f$ is a bounded function on $E$ and $f\in L^{...
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37 views

$X=\mathbb{R}^n$ is reflexive. Need help understanding the proof.

I know that every finite dimensional normed space is reflexive, so this would imply X is reflexive. However, the author writes that this also follows directly from the fact that the second dual of $...
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3answers
56 views

If $X/F$ is separable, must $X/F$ isomorphic to $F$?

Suppose $X$ is a Banach space and $F$ is a closed subspace of $X$. Clearly $X/F$ is a Banach space, equipped with quotient norm. Question: If $X/F$ is separable, must $X/F$ isomorphic to $F$? The ...
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1answer
84 views

Best approximation theorem. Hilbert space

$Y$ is a closed subspace of a Hilbert space $H$ and $x \in H$. To prove that $x \in $ orthogonal complement of Y if and only if $||x-y|| \geq ||x||$ for all $y \in Y$. Necessary condition is easy to ...
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72 views

Richardson's Methods

I need to prove Richardson's Method and the first part of the proof is: Consider the linear system $Ax = b$ where the eigenvalues of $A$ are real and positive. Let $G_{\omega } = I - \omega A$, ...
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2answers
52 views

Normed algebra with bounded multiplication

Sometimes one finds the following definition of a normed algebra: This is an algebra with a norm on the underlying vector space such that there is a constant $K \geq 0$ such that $|x \cdot y| \leq K |...
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1answer
38 views

Rearrangement of Schauder basis

My question is a part of an exercise in Banach space theory. In the space $c_0$, for $n \in \mathbb{N}$, let $s_n=\sum_{j=1}^n e_k=(1,1,\cdots,1,0,\cdots,0)$. It's easy to see that $(s_n)_{n\ge 1}$ ...
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0answers
22 views

Check uniform equicontinuity of a function family

I am struggling to prove or disprove that the following function family is uniformly equicontinuous. $$F = \{f \in C^1[0,1]: \forall x \text{ } |f(x)| + \sqrt x |f'(x)| \leq 1 \}$$ First I tried to ...
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1answer
32 views

properties of non-extreme points

I'm reading a proof of the following lemma Assume that $K$ is a compact convex set in a Hausdorff locally convex space, and $K$ is metrizable with the induced topology. Then the set $\textrm{ex}(K)...
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41 views

Linear independence and Schauder basis

It follows from the definition that a Schauder basis must be linear independent, i.e. every finite subset of the Schauder basis is linear independent. I wonder if the following "converse" of this is ...
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1answer
55 views

An inequality with a function twice differentiable

Function f: $\Bbb R \to \Bbb R$ is of Class $C^2$, Suppose $f(0)=1/2$, $f'(0)=1$, $|f''(x)| \le 1 $ $\forall x \in[-1,1]$. We also have $|f(c+k)-f(c)-f'(c)k|\le \frac{k^2}{2}$, for $c, c+k \in [-1,1]$....
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1answer
34 views

Show a function is in $L_2$

Suppose $f\colon [0,\infty) \rightarrow \mathbb R$ is measurable, and suppose there exists $M>0$ s.t. $\forall h\in\mathbb L_2[0,\infty)$ We have $ |\int _{[0,\infty)}f(x)h(x)d\mu (x)|\leq M\cdot \|...
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1answer
106 views

Weak Gâteaux Derivative

Suppose $X$ and $Y$ are Banach spaces. Let $F:X \rightarrow Y$ be a function and $U \subset X$ be an open set. The Gâteaux derivative of $F$ at $u \in U$ in the direction $\phi \in X$ is given by $$...
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1answer
15 views

Convex function outside of segment

I understand that a convex function has: $$f(tx+(1-t)y)\leq tf(x) + (1-t)f(y),\quad t\in[0,1]$$ But I don't understand why this turns around for $t\in \Bbb R\backslash[0,1]$. $$f(tx+(1-t)y)\geq tf(...
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1answer
55 views

Inequality for Sobolev fractional spaces

I recall that the Fourier transform of a function $f \in L^1 (\mathbb{R})$ is defined by $$\hat{f}(\xi) = \frac{1}{\sqrt{2 \pi}} \int_{\mathbb{R}} f(x) e^{- i x \xi} \, dx.$$ We can define that ...
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0answers
43 views

Variational formulation curl-div equation

I want to prove that the following problem admits a unique solution $A_I\in H(curl;\Omega_I)\cap H(div;\Omega_I)$. $$ \begin{cases} curl(\varepsilon_I^{-1}curl A_I)=curl v_I\;\;\text{in}\,\,\Omega_I\\ ...
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1answer
62 views

Adjoint Operator with Matrix

Let $\Omega \subset \Bbb R^n$ and $L$ be a linear differential operator 1 $$L:V\to W, V:=\mathcal C^1_0(\Omega;\Bbb R), W:=\mathcal C^0_0(\Omega; \Bbb R^{n*n})$$ $$L(u):=(\partial_j u_j - \partial_i ...
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1answer
33 views

Using Heaviside step functions to simplify integral of random variables.

I am going through Rozanov's "Introduction to probability", c4q9. The problem is set up in the following way: Two random variables $p_{\xi_1}, p_{\xi_2}$ are given by $p_{\xi_1} = \left\{ \begin{...
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1answer
411 views

Show that $G(y^*)(h) = \int \limits _X {y^* \circ F(t) \Bbb d \nu _h (t)}$ for some measure $\nu$ on $X$

The following image is taken from this paper, page $129$. Questions: Why do we have $$G(y^*)(h) = \int \limits _X {y^* \circ F(t) \Bbb d \nu _h (t)} ?$$ (I couldn't get my hands on the book '...
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3answers
81 views

Isometric copy of $\ell_1$ in $C[0,1]$?

How do I show that there is an isometric copy of $\ell_1$ in $C[0,1]$? If I obtain a sequence of functions $(f_n) \in C[0,1]$ with the following properties, then I am done: $ \|f_n\| =1 $ For all ...
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2answers
24 views

How to notice symmetry?

What is the easiest method to notice the symmetry of the following function without using any graphical tool: $$g(x)=\frac{1}{\pi \sqrt{4-x^2}}$$
1
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1answer
28 views

Is there any standard name for this theorem about extension of bounded linear operators in normed spaces without changing the norm?

Let $X$ and $Y$ be normed spaces, both real or both complex; let, in addition, $Y$ be a Banach space; let $V$ be a (vector) subspace of $X$; let $T \colon V \to Y$ be a bounded linear operator; and ...
1
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1answer
90 views

Show a function has the relationship as following

We have f: $\Bbb R \to \Bbb R$. Suppose $|f''(x)| \le 1 $ $\forall x \in[-1,1]$. Show that over [-1,1] we have: $$|f(c+h)-f(c)-f'(c)h|\le \frac{h^2}{2}$$, for $c, c+h \in [-1,1]$. I know that f is ...
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2answers
49 views

In a normed vector space X: $x_n \to x$ weakly iff $d(x_n) \to d(x) ~\forall d \in D$, $D$ dense in $X^*$

Good day, I have the following task: Let $(x_n)_{n \in \mathbb{N}}$ be a sequence in the normed vector space $(X, || \cdot || )$ and let $x \in X$. Show that the following are equivalent: (i) $x_n$ ...
4
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2answers
69 views

Hilbert Spaces and Banach Spaces

I have a problem with the definition of Hilbert Space and Banach Space. What is the difference between a Hilbert Space and a Banach Space?