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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Dual space is isometrically isomorphic

Let $\{X_i:i\in I\}$ be a collection of normed spaces. If $1\leq p < \infty$, show that the dual space of $\bigoplus_p X_i$ is isometrically isomorphic to $\bigoplus_q {X_i}^\ast$, where ...
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7 views

heat equation-calculate the temperature of an bar

let an bar of lenght 50 cm, and temperature on t=0 is 100 degree. The question is calculate the degree on the middle of the bar. So i try to write the heat equation: $\dfrac{\partial u}{\partial t}= ...
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6 views

Show the operator $S$ defined by $(Sa)_n=\bigg(\frac{3}{5}\bigg)^n a_n$ is bounded on $l^2$ and find the operator norm. Is $S$ is invertible?

Define the operator $S: l^2 \to l^2$ by $$(Sa)_n=\bigg(\frac{3}{5}\bigg)^n a_n$$ for all $n \in \mathbb{N}$ an $a_n \in \mathbb{R}$. This is how we show it is bounded. ...
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7 views

Proving a linear operator is compact: understanding the statement “norm limit of a sequence of finite rank operators”.

I am having serious trouble understanding the proof that an operator is compact. This is the original question I asked and the proof is included very helpfully in the answer. Show if $\lim_{n \to ...
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10 views

Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space

I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why $d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
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0answers
9 views

(solved)Riesz representation theorem for C(T)

Im dealing with the Riesz representation theorem to prove that de dual $C(\mathbb{T})^*$ is (isometric to) the space of complex Borel measures on $\mathbb{T}$. On the other hand I've read that the ...
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1answer
10 views

Calculating spectrum

The question is: Let $H = L^2(-\pi,\pi)$ and $[Au](x) = (1+x^3)u(x)$. Determine $\sigma(A)$. I'm reviewing for a test, so don't be concerned with "overhelping." I can determine the point spectrum ...
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0answers
11 views

Find $\alpha, \beta$ s.t. the following is minimized

Hello I would like to find $\alpha,\beta$ s.t. $$ e(\alpha,\beta) = ||\sqrt{1+\gamma^2}-\alpha-\beta\gamma||_\infty = || f_{\alpha,\beta}||_{\infty} $$ Is minimum (consider $\gamma \in [0,1)$, ...
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0answers
23 views

Normed space, inner product space, which is larger?

My textbook says normed space is inner product space unless it satisfies some parallelogram identity. In this case, we can conclude inner product space is a subset of normed space. However, norm is ...
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8 views

Strong convergence of Spectral Projection

Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded linear operators on $H$. Assume that $\{A_n\in B(H)\}_n$ strongly converges to $A$. $E^{|A|}(1,\infty)$ is a spectral projection of ...
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13 views

Limit of a sequence of functions with increasing domains

The problem I want to address is the following one. I have a sequence of functions $f_1, f_2, f_3, f_4, ...$ that live in different finite domains. To be more precise, for every natural $n$ it holds ...
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0answers
17 views

Show that $\int_{-\infty}^{\infty}f(\xi+i\eta,z_2,\ldots,z_n)e^{i[t_1(\xi+i\eta)+t_2z_2+\cdots+t_nz_n]}d\xi$ is independent of $\eta$

Show that $$\int_{-\infty}^\infty f(\xi+i\eta,z_2,\ldots,z_n) e^{i[t_1(\xi+i\eta) + t_2z_2+\cdots+t_nz_n]} \, d\xi$$ is independent of $\eta$, for arbitrary real $t_1,\cdots,t_n$ and complex ...
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0answers
19 views

Poison's equation for charge distributed on a ring

Let's consider the problem of determining the electrostatic potential given a charge distribution $\rho : U\subset \mathbb{R}^3\to \mathbb{R}$. In that case, the potential satisfies the Poisson ...
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1answer
11 views

Convex subsets of a topological vector space

I'm trying to prove: Let $X$ be a topological vector space and $A \subseteq X$. $A$ is convex if and only if $$\forall s,t \in \mathbb{R}_{+}, (s+t)A = sA + tA $$ Let $sx + ty \in sA + tA$. How ...
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0answers
19 views

Strong Solutions to Nonlinear ODE by Contraction Mapping

Consider the $1$-d ODE $$-u_{xx}+u-\epsilon u^{2}=f, \tag{1}$$ where $f$ is a nice RHS, say $f\in\mathcal{S}(\mathbb{R})$, and $\epsilon>0$. By using the Bessel potential, one looks for solutions ...
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1answer
22 views

Show if $\lim_{n \to \infty} \lambda_n=0$ then $Tu=\sum^\infty_{n=1} \lambda_n \langle u,e_n \rangle e_n$ defines a compact operator.

Let $\{e_n\}$ be an orthonormal basis in a Hilbert space $H$ and let $\{\lambda_n\}$ be a sequence of numbers. Define the operator $$T:H \to H$$ by $$Tu=\sum^\infty_{n=1} \lambda_n \langle u,e_n ...
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0answers
8 views

subset S is dense in the set of all pure states of a $C^*$-algebra with respect to the weak-$^*$ topology

Let $A$ be $C^*$-algebra and $P(A)$ the set of all states $f:A\to \mathbb{C}$ such that: for all positive $ g\in A^* $ with $g\le f$ there exists $t\in [0,1]$ such that $f=tg$. I.e. $P(A)$ is the set ...
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1answer
39 views

Do there exist bump functions with uniformly bounded derivatives? [duplicate]

Let us consider a bump function $\phi: \mathbb{R} \longrightarrow \mathbb{R}$, smooth, with compact support. The most common examples are built from the function $$ \psi(x) = \begin{cases} \exp ( ...
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1answer
26 views

Linear transformation on Banach space

Show that for $B(X,\mathbb{F})\neq \{0\},B(X,Y)$ is a Banach space if and only if $Y$ is a Banach space.($X,Y$ are normed spaces and $B(X,Y)$ = all continuous linear transformation) I proved that If ...
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2answers
111 views

Uniform unboundedness of linear operators

Question: Suppose that $(T_k)_{k=1}^{\infty}$ is a sequence of invertible linear operators on $\mathbb{R}^n$. Suppose that $\forall x \in \mathbb{R}^{n}\setminus \{0\}$, we have $$\lim_{k\to\infty} ...
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1answer
19 views

Discrete measure and piecewise function

Hi guys, can anyone please help me with why we can introduce a sectionally constant function that has support $\lambda_i, i \in \mathbb{N}$. I do not understand why we can do the part I marked with ...
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17 views

Calculate the trace of $T_nL$ where $L\in L(H)$, $T\in L(H,L(H))$ and $T_n:=\langle T,e_n\rangle_H$ for some ONB $(e_n)_n$ of a Hilbert space $H$

Let$^1$ $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space over $\mathbb R$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $T\in\mathfrak L\left(H,\mathfrak L\left(H\right)\right)$ ...
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2answers
37 views

Explicit Integrals and LimInf/LimSup

(a) Show that $f(t):=\int_0^\infty e^{-tx}\frac{sin \space x}{x}dx$ exists for $t>0$ and defines a differentiable function $f$. Calculate $f'(t)$ for $t>0$ and evaluate it explicitly. (b) Prove ...
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2answers
24 views

If a subset of metric space $(X,d)$ like $S$ is closed and bounded, does it imply that $X$ is totally bounded? [on hold]

Let $(X,d)$ be a metric space and $S$ be a subset of $X$. If $S$ is closed and bounded, does it imply that $X$ is totally bounded?
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25 views

Show that there exists $ \lambda \ge 0$ such that $v=\lambda u$

Let $\Omega \subset \mathbb{R}^n$ be open. Let $u,v \in L^1_\text{loc}(\Omega)$ with $u \ne 0$ a.e on a set of positive measure. Assume that $$\phi \in C_c^\infty(\Omega), \int u\phi > 0 \implies ...
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20 views

Why do we need to show $\|v_{mn} -w_n\| \leq \frac1m$?

My question here follows on from this question I asked a while back. If $U$ is separable and $V \subset U$, then $V$ is separable Within the proof that I wrote in the question I do not understand ...
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3answers
29 views

One of the Heire-Borel lemmas states the following:

If a set is closed and bounded in $\mathbb{R}^{n}$, then it is compact. However, what about a more abstract metric space $(X,d)$? Let $(X,d)$ be a complete metric space with a subset $A$ closed and ...
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1answer
48 views

Every normed space has a completion?

So I know that a completion of $X$ is a Banach space $Y$ such that $X$ is isometrically isomorphic to a dense subset of $Y$, say $A$. So we need to prove that we can always find a $T \in L(X,A)$ such ...
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1answer
30 views

$C_{c}^{\infty}(\Omega)$ is dense in $L^{\infty}(\Omega)$ with respect to the topology $\sigma(L^{\infty},L^{1})$

Show that $C_{c}^{\infty}(\Omega)$ is dense in $L^{\infty}(\Omega)$ with respect to the topology $\sigma(L^{\infty},L^{1})$, where $\Omega$ is an open subset of $\mathbb{R^n}$. My try: Let ...
2
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1answer
17 views

Proving function is Schwartz

I want to prove that $f(t)=e^{-t^2C\pi}$ is Schwartz. I tried computing derivatives and showing that for all $n,k\in \mathbb{N}_0$ $$\lim_{t\to \infty}t^{k}f^{(n)}(t)=0$$ but it gets messy pretty ...
2
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0answers
71 views

continuous linear functional on $l^{\infty}$ space

Let $l_{\infty}$ be the space of all bounded complex-valued sequences equipped with the supremum norm. Consider the natural standard basis $\{e_n\}_{n \in \mathbb{N}}$ of $l_{\infty}$. For any ...
2
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1answer
29 views

Operator Norm Convergence

so i am doing some exercise for my course of functional analysis. I need to show that, if $T: E \to E$ is a linear Operator and E an normed vector space over $\mathbb{C}, \mathbb{R}$ and fulfills: ...
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0answers
18 views

precompact operators in a Hilbert space [functional analysis]

I've linked to a Theorem (from H&N's Applied Functional Analysis) whose proof I'm trying to understand (I asked a question about the previous chunk of the proof yesterday). The theorem is ...
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2answers
302 views

Distance from a compact subset need not be attained in a metric space?

Suppose we have a metric space $(X,d)$. Let $S$ be a compact subset of $X$. Provide me with an example of $X$, and $S$ (closed and bounded in $X$) such that $$\min \{d(p_0,p): p \in S\}$$ does not ...
2
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1answer
36 views

Prove that if $\|A\|<1$, then $\|(I-A)^{-1}\|\geq {1\over1+\|A\|}$.

Prove that if $\|A\|<1$, then $\|(I-A)^{-1}\|\geq {1\over1+\|A\|}$. I'm not sure how to prove this result. I see feel like a geometric series is involved though. Any solutions or hints are ...
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0answers
19 views

Show $\rho_j =\sum^\infty_{k=1} \alpha_{jk} \epsilon_k$ is a compact operator

Let $\alpha_{jk}$ be numbers for $j,k \geq 1$ such that $$\sum_{j=1}^\infty \sum_{k=1}^\infty |\alpha_{jk}|^2 <\infty$$ Define $T: l^2 \to l^2$ as follows. If $x=(\epsilon_j)$, $y=(\rho_j)$ and ...
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2answers
19 views

How to show that a function in $L_2$ space is uniformly bounded in the “truncated norm” [on hold]

Let $$L_2[0, \infty) = \left\{f: \mathbb{R}_{+} \to \mathbb{R}^n \mid \int_0^\infty f(t)^Tf(t) \, dt< \infty \right\}$$ Define the truncated norm as $$\|f\|_K = \sqrt{\int_0^K f(t)^Tf(t) \, ...
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1answer
28 views

When finding the spectrum of $(Ku)(x)=\int^1 _0 k(x,y)u(y) dy$, where does $\lambda u(x)=\int^x _0 yu(y) dy + x \int^1 _x u(y) dy$ come from?

The integral operator $K:L^2 ([0,1]) \to L^2([0,1])$ is defined by $$(Ku)(x)=\int^1 _0 k(x,y)u(y) dy$$ Where $k(x,y)=min\{x,y\}$ for $0 \leq x, y \leq 1$ When finding the spectrum ok $K$ we let ...
1
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1answer
22 views

Operator groups

In $H := L^2(\mathbb{R}, \lambda)$ Hilbert-space, the following two, one-variable operator groups are given: $$(U_s f)(x):=f(x-s)$$ $$(V_s f)(x):=e^{is x} f(x)$$ $f \in H, s \in \mathbb{R}$. a, ...
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0answers
13 views

Locally Lipschitz $H^1$ and $L_{2}$

I have a doubt. Let $\Omega\subset \mathbb{R}^n$ and the function $f:H^1{\Omega}\to L_{2}(\Omega)$ defined by $f(u)=-|u|^{p-1}u$. I have to proof that $f$ is locally lipschitz if $2p<p^*$,with ...
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21 views

Graph of a higher-order function

When we deal with functions which work on numbers, we can graph them easily: Just take each of its possible input values and find its corresponding point on one axis, then go straight up in its ...
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0answers
14 views

Counterexample for the density of smooth functions in Sobolev spaces on a manifold

I'm in desperate need for help understanding a counterexample for the assertion that for (appropriate) manifolds $M,N$ the space $C^\infty (M,N)$ is dense in $L^p (M,N)$ if $\dim(M) > p$. (The ...
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21 views

Let $F$ be a field and let $F[x]$ be a vector space of polynomials $p(x) = \sum a_ix^i$ in the variable $x$ with coefficients $a_i \in F$ [on hold]

Let $F$ be a field and let $F[x]$ be a vector space of polynomials $p(x) = \sum a_ix^i$ in the variable x with coefficients $a_i \in F$. For $k\geq 1$ define $E_k(p(x)) = a_k$. Show that $E_k$ is a ...
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0answers
14 views

A question on Bounded Approximation property

Let $V$ be a Banach space and we say that $V$ has the $C$-BAP if there exists a net of bounded finite rank operators $T_\alpha$ in $B(V,V)$ and a constant $C$ such that $\|T_\alpha\| \leq C$ for each ...
4
votes
0answers
35 views

Operator continuity on Hilbert space

Let $A: H \to H$ be a linear operator on Hilbert space $H$, and let $\{\alpha_n\}_{n = 1}^{\infty} \subset \mathbb{R}$ converges to nonzero number. Prove that if the series $\sum_{n = 1}^{\infty} ...
0
votes
1answer
15 views

Does uniformly boundness in $W^{1,1}$ implies strong convergence in $L^{1}$?

Suppose $f_i$ is uniformly bounded in $W^{1,1}$. My question is, can we conclude that there exists a sub-sequence of $f_i$ convergent strongly to some $f$ in $L^{1}$? I am just reading a paper, this ...
0
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1answer
16 views

If an operator A on a Hilbert space has compact resolvent, is Ker($\lambda-A$) finite dimensional?

If an operator A on a Hilbert space has compact resolvent, is Ker($\lambda-A$) finite dimensional, for any $\lambda$ in A's spectrum? P.S: What I know now is that the spectrum of A is discrete.
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0answers
7 views

unicity solution of partial differential equation

let the following problem: $$ \begin{cases} \dfrac{\partial^2 u}{\partial t^2}= a^2 \dfrac{\partial^2 u}{\partial x^2}\\ u(0,t)=u(l,t)=0\\ u(x,0)=f(x)\\ \dfrac{\partial u}{\partial x}(x,0)=g(x) ...
2
votes
0answers
15 views

Poincare type inequality on unit square: can you improve on my constants?

I am trying to bound $\int_{[0,1]^2} u^2$ in terms of its gradient and boundary integrals, $\int_{[0,1]^2} |\nabla u|^2$, $\int_{\partial[0,1]^2} u^2$, with the best possible constants. So far I ...
3
votes
2answers
52 views

Show that $f(\phi)=\sum_{k=0}^\infty \phi^{(k)}(k)$ for $\phi \in D$ has no finite order

Let $\phi \in D:=D(K)=C^\infty_0(K) $ be a test function and let $f \in D^*=\{f:D \to \mathbb{R} : f \text{ bounded and linear} \} $ be a distribution. A distribution has finite order if: $$\exists ...