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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144 views

Do non-square infinite matrices exist?

Sorry, I tried to wrap my head more around this, but I failed. Given non-square matrix $A$ that has dimension $kn \times n$. Now let $n$ goto infinity. Is the matrix finally square?
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2answers
118 views

functional calculus and spectral measure

Let $T$ be a normal operator and $f$ be a bounded borel function on ${\sigma}(T)$. If $E_{T}$ and $E_{f(T)}$ are the spectral decompositions of $T$ and $f(T)$ respectively, prove that for any borel ...
0
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1answer
53 views

Norm of operator between two $L^p$ spaces?

In my reading, I've come across notation like $ ||T||_{2 \to \infty} $, where $T$ is an operator defined on every $L^p$ space. What does this mean? Is it simply the norm of $T$ viewed as an operator ...
2
votes
1answer
110 views

Flaw in proof that a functional is not continuous

I am trying to show Consider $F\colon C[a,b] \rightarrow R$ ; $F(f)=f(t_0)$ where $a<t_0< b$ . Show that this linear functional is not continuous under $\|\cdot\|_1$ on C[a,b] I have ...
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0answers
39 views

Characterizing elements in $X^{\ast}\hookrightarrow L^{\infty}(G)$

I am in the following situation: Let $G$ be a locally compact group (possibly Hausdorff). Let $\Phi:L^{1}(G)\twoheadrightarrow X$, where $X$ is a Banach space. Then ...
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1answer
102 views

infinity, p2 and p1 Norm and its associated unit ball that has negative y values

ok I try to phrase it in a way that is not redundant with other questions. I am surprised with the shapes that the norms take on BUT only as far as their extension into the -y regions go. How can ...
2
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1answer
65 views

completeness of a sequence space

Consider the following space: $X(P):=\{(x_n)_{n\geq0}\in\mathbb{R}^{\mathbb{N}}:(\lambda_nx_n)_{n\geq0}\in l_1, \forall (\lambda_n)_{n\geq0}\in P\}$, where $P$ is a random set of real sequences st. ...
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2answers
107 views

normed linear space of polynomials restricted to $[a, b]$

I have trouble with this problem Let $X$ be the normed linear space of polynomials restricted to $[a, b]$ . For $P \in X$, define $\phi(P)$ to be the sum of the coefficients of $P$. Show that $\phi$ ...
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2answers
78 views

old prelim exam question

In studying for my prelims, I can't quite come up with a solution to the following problem, from a few years ago. It has the feel of a "write the correct thing down, and it's 3 lines" problem. ...
1
vote
1answer
59 views

Some Help to prove $\|T^{-1}\|=\|T\|^{-1}$

I'm trying to prove that given to Banach spaces $X,Y$, and a continuous linear transformation $T:X\to Y$ with bounded inverse $T^{-1}:Y\to X$. Then, $$\|T^{-1}\|=\|T\|^{-1}$$ I already know that ...
4
votes
3answers
130 views

generalization of Banach fixed-point theorem on short maps?

If $ \ T:X \longrightarrow X \ $ is contraction, then using Banach fixed-point theorem we know that the fixed point exists and all other points converge to that point. But what happens if $T$ is not ...
4
votes
1answer
81 views

Riesz Fischer theorem?

I was wondering about the following: I read that Fischer-Riesz says that $L^2([0,1])$ is isomorphic to $l^2(\mathbb{N})$. Now it is obvious, that this should not depent on the fact which compact ...
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1answer
117 views

Is the space of continuously differentiable functions over Polish spaces Polish?

Let $(X, \|\cdot\|_X)$ and $(Y, \|\cdot\|_Y)$ be two separable Banach spaces. Consider the space of continuously differentiable functions mapping $X$ to $Y$; i.e. $C^1(X, Y)$. Consider the usual ...
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vote
1answer
99 views

Bessel function with shifted argument

Is there any standard practice which may represents $J_m(a\pm kx)$ in terms of $J_m(kx)$ where $a$ is any constant and $m$ is integer $>-1$
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1answer
33 views

Weak convergence of the 4-th degree of a weak convergent sequence

Good day! We solve an optimal control problem $$ J(u) = \|y - y_d\|^2 \to \inf $$ where $y$ is a solution of the PDE $$ \frac{dy}{dt} + Ay = Bu. $$ $A$ is a nonlinear operator, $(Bu, v) = ...
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1answer
34 views

Compactness Invariant between normed spaces

Let $X$ and $Y$ be finite dimensional normed spaces. Let $D:\X \rightarrow Y$ be an isometric isomorphism then if $X$ is compact the $Y$ is also compact. I have started by choosing a sequence in $Y$ ...
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1answer
40 views

If $f_n \to f$ uniformly a.e. and $u_n \to u$ a.e., does $f_n(u_n) \to f(u)$ a.e.?

Here $f_n$ and $f$ are real-valued functions of 1 variable. Suppose that $f_n \to f$ uniformly on $\mathbb{R}\backslash \{0\}$. Let $u_n \to u$ pointwise a.e. on a manifold $X$. Does it follow that ...
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0answers
139 views

Find features in a Signed Distance Field

I'm currently trying to improve a meshing algorithm for signed distance fields (which are of the form $f(x,y,z) = w$, where $x,y,z$ is the location of my query, $w$ indicates the distance to the ...
2
votes
2answers
84 views

What's wrong in this reasoning of $l_\infty$ separability?

While solving a problem, related to functional analysis, I've accidentally got a "proof" of $l_\infty$ being separable, tried to find a fault in it (as the result isn't true), but didn't succeed in ...
2
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0answers
41 views

$(g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s}$ right or wrong?

I am proving something, and I may need the following inequality: $$ (g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s} $$ where $(g,f)_{L^2} $ is inner product and $f$ and $g$ have higher enough regularity ...
2
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1answer
90 views

Which of these things is not like the others?

What's in a name? Well quite a lot, if you're confused enough. I have an engineering-style mathematics education, based on good old hand waving and learning bits and pieces from all over the place. I ...
4
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1answer
336 views

Zeros/poles at Laplace and at Fourier Transform

I recently started "relearning" the Laplace transform, and I noticed something. It seems to me that the intuitive idea of poles and zeros is different between these two transforms! For example, in ...
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1answer
21 views

Density of series the functions

Let $(g_n)_{n\in\mathbb{N}}\subseteq {\cal C}^0([a,b],\mathbb{R})$ with $\sum\int_a^b|g_n(t)|dt < \infty$ Prove that $\sum|g_n(t)|$ converges Show that ...
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0answers
107 views

Passing to the limit in a PDE; problem with subsequence (please check my answer)

For $w \in L^2(0,T;H^1)$, consider the PDE $$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$ where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and ...
2
votes
1answer
94 views

Weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$

What does weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$ means? $\Omega$ is open, bounded, has boundary smooth and etc...
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votes
2answers
152 views

generalized eigenspace direct sum

Similar to the way an infinite dimensional hilbert space can be written as a direct sum of eigenspaces of a normal compact operator, I was wondering whether it can be written as a direct sum of ...
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3answers
187 views

Subspace of metric space with finitely many points is complete [closed]

Show that: If a subspace $Y$ of a metric space consists of finitely many points, then $Y$ is complete.
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votes
0answers
47 views

Dual space of Sobolev functions with homogenous neumann BC

Let $\Omega$ be a bounded Lipschitz-domain with outward normal vector $\nu$ and let us take a look at the Sobolev-spaces $H^1:=H^1(\Omega)$, $H^1_0:=H^1_0(\Omega)$ and ...
2
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1answer
388 views

Functional weakly lower-semicontinuous [duplicate]

If $X$ is a topological space, then a functional $\varphi:X\to\mathbb{R}$ is lower-semicontinuous (l.s.c) if $\varphi^{-1}(a,\infty)$ is open in $X$ for any $a\in\mathbb{R}$. If $X$ is a Hilbert ...
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0answers
48 views

Self-Adjointness of beam stiffness operator

I think it is well known that the operator $\frac{EI}{\rho} \frac{\partial^4}{\partial x^4}$ which arises from a standard Euler-Bernoulli beam is self-adjoint in $H$, where $H = L^{2}$, given ...
1
vote
1answer
63 views

Check continuity and find norm of linear functional

I have to check continuity and find norm of the following linear functional $$\ell^{2}\ni \{x_n\}^{\infty}_{n=0} \rightarrow \sum^\infty_{n=0}\frac{x_n}{\sqrt{2^n}}\in \mathbb{K}$$ As for continuity, ...
0
votes
1answer
38 views

Showing that if there exist isometries $S_1,S_2 \in L(V)$ such that $T_1 = S_1T_2S_2$, then $T_1$ and $T_2$ have the same singular values.

Suppose that $T_1,T_2 \in L(V)$. I would like to show that if there exist isometries $S_1,S_2 \in L(V)$ such that $T_1 = S_1T_2S_2$, then $T_1$ and $T_2$ have the same singular values. The proof: ...
0
votes
1answer
77 views

Completeness preserved under isometric isomorphism

Let $X$ and $Y$ be finite dimensional normed spaces. Let $D: X \rightarrow Y$ be an isometric isomorphism. If $X$ is complete is $Y$ complete? Any hints on how to start this would be great!
4
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1answer
47 views

What about $\ell^1$ with pointwise multiplication

This question occurred to me after reading this thread. I was working on finding an example of a Banach algebra. The example I came up with was $\ell^1 (\mathbb N)$ with pointwise multiplication. I ...
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2answers
39 views

How can I verify that $\langle a_1e_1+\cdots+a_ne_n, b_1e_1+\cdots+b_ne_n\rangle = a_1b_1+\cdots+a_nb_n$ is an inner product?

In Sheldon Axler's Linear Algebra done right, he says that if we let $U$ be a finite dimensional real vector space and let $T \in U$, then an inner product on $U$ is $\langle a_1e_1+\cdots+a_ne_n, ...
1
vote
1answer
21 views

w*-convergence vs. convergence on a dense subspace

Let us have a Banach space $X$, a dense subspace $D\subseteq X$, a net $\{\phi_{i}\colon i\in\mathcal I\}$ in $X^*$ and $\phi\in X^*$. Suppose that $$\lim\limits_{i\in\mathcal I}\phi_{i}(d)=\phi(d)$$ ...
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1answer
244 views

Normed linear space with two norms that are not equivalent, one is complete, what about the other?

I have been searching for an answer to the following question: Given a normed linear space $V$ and two norms that are not equivalent, but $\exists K\in\mathbf{R}$ such that $\|v\|_1\leq K\|v\|_2$ ...
5
votes
1answer
96 views

Closure of compact sets in Banach space

Let $(X,\vert\vert\cdot\vert\vert)$ be a Banach space. For each $k\in\mathbb{N}$ let $A_k\subseteq X$ be compact and $r_k\in\mathbb{R},r_k>0$, such that $$A_{k+1}\subseteq \{x+u\vert x\in A_k ...
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vote
1answer
101 views

Is the expected value continuous in some sense?

For example if we have the space of random variables $L^1$. Then we should have that $|E(X-Y)| \le ||X-Y||_{L^1}$, right?- So, this would mean, that the expected value is Lipschitz continuous, ...
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1answer
76 views

Densely injective bounded linear map that is not injective

Suppose $T:X\rightarrow Y$ is a continuous linear map of Banach spaces, say. Let $D$ be a dense subspace of $X$ and assume $T$ is injective on $D$. Does it follow that $T$ is injective? I would ...
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1answer
57 views

Continuous functional that separate points

This is an exercise from Royden's Real Analysis. Let $X$ be a normed linear space and $W$ a subspace of $X^*$ that separate points. For any topological space $Z$, show that a mapping $f:Z\to X$ ...
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1answer
24 views

Identify each $x_n$ with a function $L_n \in (\mathcal{l}^1)^*$

Let $x_n=(0,...,0,1,..1,..)$ be a sequence whose first n components are all equal 0 and the rest of the terms are equal 1. Identify each $x_n$ with a function $L_n \in (\mathcal{l}^1)^*$, and then ...
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2answers
83 views

Show that $(\mathcal{l}^\infty)^*$ is not homeomorphic to $\mathcal{l}^1$

Show that $(\mathcal{l}^\infty)^*$ is not homeomorphic to $\mathcal{l}^1$, by showing that the dual of $\mathcal{l}^\infty$ contains a proper subspace which is homemorphic to $\mathcal{l}^1$. L is ...
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2answers
60 views

Show that a map is a continuous bilinear form on $H^1(0,1)$ space

Let $u,v \in H^1(0,1) = \{f : (0,1) \longrightarrow \mathbb{R}, f,f' \in L^2(0,1) \}$, show that $$a(u,v) = \int_0^1 (u'v' + uv)\; dx$$ is a continuous bilinear form.
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1answer
48 views

The norm of operator matrix

Let $H$ be a Hilbert space and $B(H)$ be the bounded linear operator on $H$, for $T\in B(H)$, if $T=\left(\begin{array}{ccc} 0 & B \\ A & 0 \\ \end{array}\right)$ on $H=M\oplus ...
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1answer
50 views

The convergent in the point-ultraweak topology

Let $A$ be a C*-algebra , $B(H)$ be the bounded linear operator on Hilbert space $H$ and $P_{i}\in B(H)$ be an increasing net of finite-rank projections which converge to the identity in the strong ...
0
votes
1answer
61 views

Does a clean proof exist of why $\mbox {dim range}T=\mbox {dim range}T^*$?

Does a clean proof exist of why $\mbox {dim range}T=\mbox {dim range}T^*$? Here T^* is the adjoint or the conjugate transpose. This is assuming we have a finite dimensional vector space such that $T ...
0
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1answer
37 views

What allows us to take the adjoint of both sides of an equation?

Suppose that I have that $ST = TS =I$ Where $T,S \in L(V)$ What allows me to take the adjoint of all three sides to get: $T^*S^*=S^*T^*=I^*=I$
2
votes
1answer
79 views

Dual of Hilbert space dense in dual of Reflexive space.

I don't see how to solve this problem which I think should be easy: Let Y be a reflexive space. Assume $Y$ is continuously embedded in a Hilbert space $H$ and $Y$ is dense in $H$. Show that $H^*$ is ...
1
vote
1answer
45 views

spectral measure and integral query

I have proved the 'resolution of the identity' for a normal operator, namely that there is a unique spectral measure E such that $\int_{{\sigma}(T)} {\lambda}\,dE=T$ If (${\lambda}_{n}$) is the ...