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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5 views

Dual map is zero if and only if map is zero

A problem from Linear Algebra Done Right (Third Ed): Suppose $W$ is finite dimensional and $T \in \mathcal{L}(V, W)$. Prove that $T=0$ if and only if its dual $T' \in \mathcal{L}(W', V')=0$. I am ...
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1answer
16 views

Computation of an integral depending on the Legendre polynomials

Let $P_l$ be a Legendre polynomial ($l$ is an integer). I want to know why the quantity $$ v_l(k):=(-i)^l\int_{-1}^{+1}\mathrm{e}^{ikx}\,P_l(x)\;\mathrm{d}x $$ is real?
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1answer
21 views

Completeness and orthogonal projection

a. Which are the properties that define an orthogonal projection? Give a precise definition. b. What does completeness mean? Please state both the definition and an example (without proof) ...
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0answers
11 views

Find an operator on $C[0,1]$ with a given compact set in $C$,the complex field

Let $K$ be a non-empty compact subset of $C$,the complex field. Does there exist an operator in $\mathscr{B}(C[0,1])$ such that $\sigma(A)=K$? $A$ is a multiplication operator iff $K$ is non-empty ...
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13 views

Symbol for continuous dual

Let $X$ be a vector space. The dual is the set of all linear forms from $X$ to $\mathbb{F}$.(can be $\mathbb{R}$ or $\mathbb{C}$) That is, $X^* = \operatorname{Hom}(X,\mathbb{F})$. If $X$ is also ...
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0answers
11 views

Existence of nuclear dominating positive definite kernel

Let $\mathcal{X}$ be a metric space and $k: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$ a continuous positive definite kernel. Can we always find a positive definite kernel $r$ such that $r \gg k$ ...
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2answers
23 views

Is a self-adjoint operator continuous on its domain?

Let $H$ be a Hilbert space, and $A : D(A) \subset H \rightarrow H$ be an unbounded linear operator, with a domain $D(A)$ being dense in H. We assume that $A$ is self-adjoint, that is $A^*=A$. Since ...
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0answers
10 views

The convergence rate of the derivative of a sequence of function

Let $v_\delta$ be a sequence of continuous diff'able function on $(-1,1)$ and $0\leq v_\delta\leq 1$. For each $\delta>0$, assume that $v_\delta(\delta)=v_\delta(-\delta)=1$ and $v_\delta(0)=0$. We ...
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0answers
11 views

An example of convergence to Young measures

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\lam}{\lambda}$ I am trying to prove the following claim: Let $\{u:[0,1]\to \mathbb{R} \mid u \, \, \text{ is differentiable a.e}, u(0)=u(1)=0 \}^{*} $. ...
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1answer
32 views

Why isn't every Hamel basis a Schauder basis?

I seem to have tripped on the common Hamel/Schauder confusion. If $X$ is any vector space (not necessarily finite dimension) and $B$ is a linearly independent subset that spans $X$, then $B$ is a ...
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26 views

Differentiation as Rotation

I am trying to make a connection between linear algebra and the Fourier transform. Functions form a vector space and differentiation is an operator. Fourier transforming a function from what i ...
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1answer
44 views

Show that $|T(x) - T(y)| \lt |x-y|$ when $x \neq y$ but the mapping has no fixed points.

Consider $X = \{ x \in \mathbb{R} \mid 1 \le x \lt \infty \}$, taken with the usual metric of the real line, and $T \colon X \to X$ defined by $x \mapsto x +x^{-1}$. Show that $|T(x) - T(y)| \lt ...
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1answer
19 views

Triangle inequality with a twist

Assume $t>0$ and $x,y,z\in [0,t)$ how would one go about showing $$\min \{|x-y|,t-|x-y|\}\leq\min \{|x-z|,t-|x-z|\}+\min \{|z-y|,t-|z-y|\} $$ If the first one materializes from every minimum, then ...
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1answer
21 views

Complete set in $L^2(\mathbb{R}^3)$ from Spherical Harmonics

The Spherical Harmonics form a complete set of functions on the sphere $S^2$, so that any function of $f: S^2\to \mathbb{R}$ can be written uniquely as $$f(\theta,\phi)=\sum_{l=0}^\infty ...
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1answer
19 views

How can the inverse of an operator between Hilbert spaces H,K be defined on the dual of H?

I need some help to understand the following statement. Let $A$ be an operator defined as follows: $Av = -\Delta v - \nabla \text{div} u$ It is known that the operator $A$ is positive self-adjoint ...
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1answer
23 views

Doesn't this $L^p$ norm estimate for all $p$ give me an $L^\infty$ bound?

Let $r_n \to \infty$ as $n \to \infty$. We have that $$\lVert v \rVert_{L^{r_n}(\Omega)} \leq C\lVert v \rVert_{L^{r_0}(\Omega)} < \infty$$ for all $n$, where $C$ is independent of $v$ and ...
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0answers
15 views

Prove that the standard orthonormal sequence $(e_n)^\infty_{1}$ is complete in $l^2$. [on hold]

Prove that the standard orthogonal sequence $(e_n)^{\infty}_{1}$ is complete in $l^{2}$. Where $(e_{n})$ is the sequence with nth component equal to 1 and all others zero.
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12 views

Prove that $\left\|f_n(x)-g_n(x)\right\|^2 = \|f_n(x)\|^2+ \|g_n(x) \|^2-2\operatorname{Re}\int_{\mathbb R} f_n(x)\overline{g_m(x)} dx$

Let $\{f_n(x)\}_{n\in\mathbb Z}$, $\{g_n(x)\}_{n\in\mathbb Z}$ be two sequence of square-integrable functions: $f_n, g_n\in L^2(\mathbb R)$. Prove that $$\left\|f_n(x)-g_n(x)\right\|^2_{L^2(\mathbb ...
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1answer
14 views

Dissipativity for Hilbert spaces

I want to prove that an operator $A:D(A)\to X$ is dissipative $\iff$ $\text{Re}\langle Ax,x\rangle\le 0$ $\forall x\in D(A)$. The proof for this is actually sketched on the Wikipedia page for ...
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1answer
32 views

Is every topological space is measurable?

Actually I am learning about measure theory. But I have confusion between topological space and measurable . Is there any relationship among them or not?
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13 views

prove that a non constant linear functional from a normed linear space X is discontinuous if and only if Z(f) = {x in X | f(x) = 0} is dense in X. [duplicate]

I could prove that if Z(f) is dense in X then the functional must me discontinuous but I am not able to prove the other way round
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1answer
12 views

The w*-extension of a bounded linear functional

Let Y be a Banach space and assume that $X$ is a $w^*$-closed subspace of $Y^*$. Let $f$ be a bounded linear functional on $X$. Does there exist any $w^*$-continuous linear functional $\phi$ on $Y^*$ ...
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1answer
17 views

An example of smooth compactly supported function with non-vanishing Fourier transform

I am trying to find an explicit example of smooth real-valued compactly supported function $u$ in $\mathbb R^n$, $n \geq 2$, such that its Fourier transform $\widehat u$ does not have zeros. By the ...
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0answers
35 views

Fourier Series in Functional analysis [on hold]

Would you please solve this question? I really have problem with this kind of questions.
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0answers
33 views

How to view Stone-Cech compactification of the real line?

I am going through Arveson's A Short Course on Spectral Theory and have come across an exercise constructing $\beta\mathbb{R}$ using the Gelfand map. I was wondering if there is an explicit ...
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1answer
29 views

Is $W^{1,2}_0$ a Hilbert space?

I came across the term $W^{1,2}_0$. Just a quick question on what the 0 means and is it a Hilbert space? I know $W^{1,2}$ is a Hilbert space. Thanks!
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44 views

On the definition of commuting self adjoint operators.

I'm reading Mathematical Methods in Quantum Mechanics by Gerald Teschl and I came across the following exercise whose statement is causing me some troubles. It goes like this: Let $A$ and $B$ two ...
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0answers
15 views

Showing coercivity of the bilinear form associated with a robin boundary value problem

I'm trying to show the existence and uniqueness of weak solutions to the following boundary value problem: \begin{align} -\nabla \cdot ( k \nabla u) &= f \quad \text{in } \Omega \subset ...
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1answer
19 views

Find the strong form of a PDE from the weak form.

I'm having a little difficulty understanding how to find the strong form of a PDE given the weak form. For example, I have the weak form as: $\displaystyle\int_\Omega [a(x)\nabla u\cdot\nabla ...
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2answers
53 views

Why to introduce norms of vectors?

I am studing Euclidean, metric and normed spaces. What I don't get it is why should I norm a vector. It is usually squared? Why should it be always positive? I've asked this to many people and nobody ...
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0answers
21 views

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable for $p>0$

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable if $p>0$. I did show it is a bounded set because if there exists $x^{(N)}\subset A $ such that $||x^{(N)}||\to \infty $ ...
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1answer
27 views

If $\|\cdot\|_{1}\le\|\cdot\|_{2}$ then $\|\cdot\|_{2}\le M\|\cdot\|_{1}$ [duplicate]

Let $(X,\|\cdot\|_{1})$ and $(X,\|\cdot\|_{2})$ be complete normed vector spaces and $\|x\|_{1}\le\|x\|_{2}$ $\forall x\in X$. I want to prove that $\exists M>0$ such that $\|x\|_{2}\le ...
4
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5answers
61 views

About inner products, norms and metrics

Do these three kinds of vector spaces, those with an inner-product, those with a norm and those with a metric, are the same sets of vector spaces? At least for finite dimensional vector spaces all of ...
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1answer
65 views

Every Hilbert space is isometrically isomorphic with $\ell^2$

Let $H$ be a hilbert space and let $\{u_\alpha\}_{\alpha \in A}$ be a orthornormal basis ($A$ is not supposed to be countable a priori). Then there is an isometric isomorphism between $H$ and ...
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1answer
37 views

How to define properly the radial and angular dependence of a function?

Recently I've came across the following situation when studying Quantum Mechanics: suppose we have two operators $A,B$ on the space of functions $L^2(\mathbb{R}^3)$ and suppose they have the following ...
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1answer
27 views

How does the general spectral theorem generalize the simpler versions?

I read the following version of the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas: I'm trying to understand why this is a generalization of the following version, ...
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2answers
51 views

Why can entire function be written as exponential, and why is it bounded in this way?

Let $A$ be a commutative complex Banach algebra with unit element $e$. Suppose now that $f(x) \in \sigma(x)$ for every $x \in A$ where $\sigma(x)$ denotes the spectrum of $x$. Now, let $x\in A$ and ...
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0answers
36 views

The closure of the closed ball is a closed [on hold]

In how many ways you can show that $\overline{\overline{B}(x,r)}=\overline{B}(x,r)$ where $\overline{B}(x,r)=\lbrace y \in \mathbb{R}^n : d_e(x,y) \leq r \rbrace$, and $d_e$ is the euclidean metric ? ...
2
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1answer
18 views

Sequential compactness of smooth functions

Suppose I have a sequence $u_n$ of smooth functions on the $N$-dimensional reals. If $\|D^{\alpha}u_n\|_{\infty} \leq C_{\alpha}$ for all multi-indices $\alpha$, then is it possible to deduce that ...
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1answer
33 views

Why is $f$ continuous if its kernel is not dense in $A$?

Let $A$ be a commutative complex Banach algebra with unit element $e$. Suppose $X \subset A$ has codimension $1$ and consists out of non-invertible elements. Clearly $X$ is the kernel for some ...
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57 views

Is the sum of infinitely many open sets open?

Let $X$ be a locally convex space (or, in particular, a normed space). Let $(O_n)_{n=1}^\infty$ be an infinite sequence of non-empty open sets in $X$ such that the sum $\displaystyle\sum_{n=1}^\infty ...
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1answer
21 views

Reference request: Every vector subspace of codimension $1$ of a vector space $A$, is the kernel of some nonzero functional $f$.

I know the following statement is true, but I am looking to find a good reference that proves this quite nicely Every vector subspace of codimension $1$ of a vector space $A$, is the kernel of ...
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1answer
24 views

Is this element of $H^1(\Omega)^*$ actually in $L^2(\Omega)$?

Let $\Omega$ be a smooth bounded domain. Let $v \in H^2(\Omega)$ satisfy $-\Delta v = 0$ on $\Omega$ with $\partial_\nu v = g$ where $g \in H^{1/2}(\partial\Omega)$ is normal derivative data. ...
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0answers
21 views

Galerkin method in Sobolev space

I've got this problem to solve: Using Galerkin method, prove that there exists a weak solution of this differential equation: $$-\Delta u = a(x) \circ \triangledown u - u_t +f(x)$$ on $\Omega$ $$u ...
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1answer
38 views

How to write this down in a rigorous manner?

I'm studying Quantum Mechanics and there's something I'm in doubt in how to write it down in a rigorous way. The idea is: consider a Hilbert space $\mathcal{H}$ and one hermitian operator $A\in ...
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1answer
21 views

Compact operators on $L_p$

Consider $M_n(\mathbb{C})$ as $B(\ell_p^n)$ for $n\in\mathbb{N}$ where $p\in[1,\infty)$, and include $M_n(\mathbb{C})$ in $M_{n+1}(\mathbb{C})$ as the upper left corner. Is it true that ...
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1answer
8 views

Does continuity in one variable and locally Lipschitz in another imply uniformity in the first?

I understand the definition of Lipschitz functions when talking of functions of single variables. However, I have trouble understanding it when it is a multivariable function. Suppose $ f(t,x):D ...
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1answer
22 views

Show that $\phi_{y,\epsilon} \in D(\mathbb{R^m})$

For a fixed $y$ $$\phi_{y,\epsilon}(x)= \left\{ \begin{array}{ll} \exp(-\frac{\epsilon^2}{\epsilon^2-|x-y|^2}) & \mbox{if $|x-y| \lt \epsilon$};\\ 0 & ...
0
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1answer
44 views

Convergent Bounded Linear Maps

I'm not sure how to show that the composition of two convergent bounded linear maps converges to the composition of their limits. I've shown that the composition of bounded linear maps is a bounded ...
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3answers
127 views

How do I get $\|x\|\le C\|y\|$ in this case?

I feel that the title is a bit uninformative, please feel free to edit it. This is a problem related to the Open Mapping Theorem. Let $T:X\to Y$ be a bounded linear operator from a Banach space X to ...