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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Compact sets are bounded: shape of the cover matters?

To prove a compact sets is bounded, we assume there's a "open ball cover" (each with R=1) that covers the set. And take maximum distance over the center of the balls +2 as the boundary. Why could we ...
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1answer
26 views

Continuous linear operator T at a point T is then continued

The problem is the next If T is continuous at a single point, it is continuous, without using that T is continuous iff T is bounded. I tried this result as follows If T is continuous at a single ...
3
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1answer
30 views

Strict positiveness on a C*-algebra given by generators and relations.

Let $A$ be a C*-algebra with generators $a_1,a_2,\ldots,a_n$ and some (non-important) relations (the relations imply that $\|a_i\|\leq 1$, so that $A$ exists). Among the given relations we have that ...
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0answers
32 views

A convergence problem for a sequence of functions

Let $\{f_n\}$ be a sequence of functions in $L^p$ with $0<p<1$ let $b>1$ and $(b^n(f_{n+1}-f_{n})\rightarrow 0)$ what we can say about $\{f_n\}$ ?is it Cauchy?
3
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1answer
32 views

Estimate for a weak solution to a PDE

Let $f \in L^2(B_R(0))$ and let $u \in W^{1,2}(B_R(0))$ be a weak solution of the equation $$Lu = - \sum_{i,j=1}^{n} D_i(a_{ij}D_ju)+ \sum_{i=1}^{n} b_i D_i u + cu =f.$$ There are constants $0 \le ...
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1answer
17 views

Codimension 1 closed subspace as a kernel of a functional

My non-linear analysis book says that if I have a linear operator $T:X\to Y$ with close range $R$ and $\operatorname{codim}(R)=1$ (and also $\dim(\ker(T))=1$) then there exists $\phi\in Y^{*}$ such ...
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17 views

Why does $M$ have a limit rank in the operator norm? Why is $S$ bounded? [on hold]

Define operator $S$ and $M$ on $\ell^2$ by $(SX)_n = \begin{cases} 0 & n = 0 \\ x_{n - 1} & n > 1 \end{cases}$ $(Mx)_n = \dfrac{1}{n + 1} x_n,\qquad n \ge 0$ Why does $M$ have a limit ...
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1answer
22 views

Sobolev space on a closed subset

Hi I have a question about Sobolev spaces. Let $U \subset \mathbb{R}^{d} $ be a bounded open subset and $dx$ be a Lebesgue measure on $U$ \begin{align} W^{1,2}(U):=\left\{u \in L^{2}(U;dx): ...
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1answer
15 views

Infinite sum of bounded linear operators on a Hilbert space

Let $\mathcal{H}$ be an infinite-dimensional, separable, complex Hilbert space, and let $\mathbf{a}$ and $\mathbf{b}$ be bounded linear operators on $\mathcal{H}$ such that ...
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2answers
24 views

Locally boundedness of some L^p spaces.

It is well known that $L^p$ spaces for $ 0<p<1 $ are not locally convex. I would like to know whether they are locally bounded.
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1answer
36 views

Show that the space $ℓ^0=\{\{a_j\}_{j=1}^{\infty}\subset \mathbb C\mid a_j=0\text{ for } j>>1\}$ is not complete

Show that the space $$\ell^0=\{\{a_j\}_{j=1}^{\infty}\subset \mathbb C\mid a_j=0 \text{ for } j\gg1\}$$ with inner product $$(a,b) \in ℓ^0\timesℓ^0 \mapsto \langle a,b\rangle =\sum_{j=1}^\infty ...
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1answer
22 views

Prob. 3, Sec. 4.2 in Erwin Kreyszig's Functional Analysis: How to show that $\lim\sup$ is sublinear?

Let's consider the real space $\ell^\infty$ of all bounded sequences of real numbers. Let $p \colon \ell^\infty \to \mathbb{R}$ be defined by $$p(x) \colon= \lim\sup_{n \to \infty} \xi_n \ \mbox{ for ...
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15 views

Domain of closed unbounded operator

Let $A$, $B$ be two closed unbounded operators such that: (1) there exists dense subspace $\mathcal{D}$ of $Dom(B)$ which is contained in $Dom(A)$, (2) for every $\psi \in\mathcal{D}$ it holds $$ ...
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2answers
43 views

Differentiating $f(x)=\sum_{i=1}^{N}|x-y_i|^2$ where $y_1,…,y_N\in \Bbb{R}^n$.

Let $y_1,...,y_N\in \Bbb{R}^n$ and let $f(x)=\sum_{i=1}^{N}|x-y_i|^2$. I need to show that $f$ has a minimum. I try to differentiate but I am having troubles doing so. First of all, does $|x-y_i|$ ...
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2answers
29 views

Functional Analysis (Topological and Isometric Isomorphisms)

Give an example that if two normed linear spaces are topologically isomorphic then they need not be isometrically isomorphic. I searched my book and on the Internet but in vain.
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1answer
58 views

In the Hahn-Banach theorem, what is the purpose of the 'dominating function'?

I am studying functional analysis by reading "Elements of Functional Analysis" by IJ Maddox (which was the set text for the Open University's now discontinued course on this subject). In the ...
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1answer
25 views

Existence of the continuous spectrum of a possibly-unbounded, linear self-adjoint operator on a complex Hilbert space

Let $\mathbf{A}$ be a possibly-unbounded, linear self-adjoint operator on an infinte-dimensional, complex separable Hilbert space $\mathcal{H}$, and suppose we know the matrix elements $\langle ...
2
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1answer
26 views

The 1-Norm on a Quantum Group as a Supremum

To this MO question, Yemon Choi comments that If $\tau$ is a faithful normal trace on a von Neumann algebra $M$, then IIRC $\tau(|x|)$ is equal to the supremum of $|\tau(xy)|$ as $y$ runs over all ...
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1answer
16 views

application of positive linear functionl

The space $L^{1}(\mathbb R)$ is a commutative Banach algebra under convolution. A linear functional $F$ on $L^{1}(\mathbb R)$ is positive if $F(f^*f)\geq 0$ for $f\in L^{1}(\mathbb R).$ (where ...
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0answers
7 views

Isometry and topological isomorphism produces a equivalence relation on the collection of all normed linear space over some field K

The concept of Isometry and topological isomorphism produces a equivalence relation on the collection of all normed linear space over some field K . I want to know how to proceed... Help needed
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6 views

Equivalence relation in the collection of all normed linear space over some field K.

The concept of equivalent norm produce an equivalence relation in the collection of all normed linear space over some field K. I've no idea how to make a start... Please help
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1answer
34 views

$C^*\!$-algebra-normal element, self-adjoint element and spectrum [on hold]

Let $A$ be a $C^*\!$-algebra. Suppose $x$ is a normal element of $A$ and $\operatorname{spect}(x)$ lies in $\mathbb{R}$. Prove that $x$ is self-adjoint.
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21 views

If for every $a > 0$, $u \in C^\infty([a,\infty))$, then is $u \in C^\infty((0,\infty))$?

Suppose that for every $a > 0$, $u \in C^\infty([a,\infty))$. Does this imply that $u \in C^\infty((0,\infty))$? I think it is true when we just work in $C^0$, but with $C^\infty$ you need to ...
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0answers
37 views

Functional Analysis (Normed Linear Spaces)

Since every normed linear space is a vector space but every vector space is not necessarily a normed linear space. Give an example to show that a vector space is not a normed linear space that is norm ...
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1answer
30 views

$L_{P}[0,1]$ space for $0<P<1$ is metric space. [on hold]

For $0<P<1$, let $L_{P}[0,1]$ be the set of measurable functions $f : [0,1]\rightarrow R$ such that $\int{|f(x)|}^{p}dx<\infty$. How the function $d(f; g) =\int{|f(x)-g(x)|}^{p}dx$ is a ...
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1answer
23 views

$f(x,y,z)=ax+by+cz$. If $\mathbb R^3$ equipped with sup norm is f be bounded? If so find $\Vert f\Vert.$

It's very easy to see $f$ is bounded with respect to 2-norm which I've already done. $$|f(x,y,z)|\leq|a||x|+|b||y|+|c||z|$$ $$\leq\sqrt{a^2+b^2+c^2}\Vert(x, y, z)\Vert.$$ Then $\Vert ...
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1answer
19 views

Series Test for Integrability via the Distribition Function

I imagine that the following question has a well known (and perhaps, easily obtainable) answer, but I can't find it by myself nor along the references that I have in mind so far. So, if $f$ is a ...
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1answer
29 views

Integral operator

Let T: $C[0,1]\rightarrow C[0,1]$ be defined by $y(t)=\int_{0}^{t}x(\tau)d\tau$. Find Img(T). I know that Img(T)={$w\in C[0,1]:w=(Ty)(t) \text{ for some } t\in C[0,1]$}. Could you give me any ...
2
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1answer
53 views

Is the supremum norm the only $ C^{*} $-norm on $ {C_{c}}(X) $, equipped with the usual pointwise operations?

Let $ X $ be a locally compact Hausdorff space. Then $ {C_{c}}(X) $ is a commutative $ * $-algebra with respect to addition, multiplication, scalar multiplication and conjugation (all pointwise ...
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0answers
21 views

If $B$ is the closed unit ball of $X$, Why does $\varphi (B)$ is $\sigma $-dense in the closed unit ball of ${X^{**}}$?

Let $\varphi $ be the embedding of $X$ into ${X^{**}}$ . Let $\tau $ be the weak topology of $X$, and let $\sigma $ be the $weak^*$-topology of ${X^{**}}$--the one induced by $X^*$. If $B$ is the ...
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1answer
20 views

When the Multiplier algebra of a Banach algebra is exactly equal to the operator algebra?

Let A be a Banach algebra. B(A) and M(A) be the operator algebra and the multiplier algebra of A, respectively. When we have M(A)=B(A)?
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21 views

Banach space and Hamel Basis cardinality

No infinite-dimensional normed linear space with a Hamel basis having cardinality strictly less than c can be complete. Can we prove it without using AC or Hahn-Banach Theorem?
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Equivalence of norms in $C^1[0,1]$

i have the following problem/questions: I have to prove that $\lVert \cdot \rVert_1 \sim \lVert \cdot \rVert_{*} $ in $C^1[0,1]$; Where $\lVert \cdot \rVert_1$ is the usual $C^1[0,1]$ norm and ...
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0answers
23 views

Part of Lomonosov's Invariant Subspace Theorem

Let $X$ be a complex Banach space of infinite dimension, let $T\in\mathcal{B}(X)\backslash\{0\}$ be compact. Define $$\Gamma := \{S\in\mathcal{B}(X)\,|\,S\circ T=T\circ S\}$$and define, for each ...
2
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1answer
47 views

A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$

A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$. How will we prove the converse implication. One sided implication for Hilbert Space is proved in ...
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1answer
37 views

Basis for $l^{\infty}$

As the question stated, we know that $\{e_i\}$ doesn't form a basis for $l^{\infty}$. So how can we find a basis for $l^{\infty}$, no matter it is Schauder or Hamel basis.
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1answer
28 views

The Legendre-Fenchel transform of $BV$ semi-norm

I am reading a numerical paper in which it calls some "easy" facts from convex analysis but I can't justify it... Let $\Omega\subset \mathbb R^2$ be open. Define for a function $u\in L^1(\Omega)$ and ...
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1answer
39 views

existence of functionals

Let $X$ be a finite-dimensional normed space. Consider a non-empty convex set $C\subset X$ such that $0\notin C$. Notice that $C$ has a dense and countable subset $\{x_n\}$. $\forall n $ let $C_n= ...
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0answers
16 views

Gauss-Legendre quadrature error

I'm trying to evaluate the error in Gauss-Legendre quadrature formulae on $[a,b]$. So far I have that the error is less or equal to $$ \frac{f^{(2n)}(\xi)}{(2n)!}\langle p_n,p_n \rangle, \enspace ...
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1answer
28 views

If $S:(x_0,x_1,x_2,\ldots) \to (0,x_0,x_1,x_2,\ldots)$ Why does $0 \in \sigma (S)$? and what is $\dim(R(S))$ in $\ell^2$?

Consider the unilateral shift $S$ on $\ell^2$ defined by $$(x_0,x_1,x_2,\ldots) \to (0,x_0,x_1,x_2,\ldots)$$ Why dose $0$ is eigenvalue of $S$? and what is $\dim(R(S))$ in $\ell^2$?( Range of $S$)
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1answer
74 views

Hamel Basis in Infinite dimensional Banach Space without Baire Category Theorem

Prove that every Hamel basis in an infinite dimensional Banach Space is uncountable without using Baire Category Theory. We are assuming axiom that vector space dimension (if exist) is well-defined. ...
2
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0answers
20 views

Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0 $$ Then, to find ''bound states'', you solve on the right and find the ...
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1answer
26 views

$L^2(a_1,b_1;H_0^1(a_2,b_2))\subset L^2(a_1,b_1;L^2(a_2,b_2))$ and Convergence

Let $[a_1,b_1]\times[a_2,b_2]\subset\mathbb{R}^2$. Suppose $$u_n\rightharpoonup u\,\,\,\text{ weakly in } L^2(a_1,b_1;L^2(a_2,b_2))$$ and $$\{u_n\}\text{ is bounded in }L^2(a_1,b_1;H_0^1(a_2,b_2)).$$ ...
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1answer
16 views

Calculate the asymptotic growth of a sum that contains log or binom

I'm looking for a basic explanation how to calculate the asymptotic growth of sums. Take for example this one: $\sum_{i=1}^{lg(n!)} 2^{n^2}$ or this one: $\sum_{i=0}^{n} {n\choose{i}}$ The ...
2
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0answers
24 views

Eigen function of one Stochastic Process from the eigen function of another Stochastic Process

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are ...
2
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1answer
47 views

Can every vector space (over $\mathbb{R}$ or $\mathbb{C}$) can be a Banach space (or Hilbert space)?

For a vector space $V$ over $\mathbb{R}$ (or $\mathbb{C}$) with Hamel basis of cardinality $\kappa$ such that $\kappa^{\aleph_0} = \kappa$, can we define inner product(or norm) on $V$ such that $V$ is ...
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votes
1answer
89 views

Is $T$ a compact mapping from $W_{0}^{1,2}\left(\Omega\right)$ into itself? [on hold]

Let $\Omega$ be an open bounded subset in $\mathbb{R}^{6}$ and $f$ be in $L^{8}\left(\Omega\right)$. For any $w$ in $W_{0}^{1,2}\left(\Omega\right)$, define $T\left(w\right)$ be in ...
0
votes
1answer
26 views

How to justify $\lVert \sum_{j=n+1}^\infty a_jh_j\rVert^2 \leq \sum_{j=n+1}^\infty a_j^2$ when $h_j$ are orthonormal

We work in a Hilbert space $H$. I want to show that a series $\sum_{j=1}^\infty a_jh_j$ converges where $h_j$ is an orthonormal basis of $H$. To do this, I want to show that the tail $$\lVert ...
0
votes
1answer
25 views

Hilbert space and uncountable cardinal

Given an uncountable cardinal does there exist Hilbert space with orthonormal basis of that cardinality?
5
votes
1answer
61 views

Equivalent formulations: pure contraction

I want to prove the following equivalence: let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. TFAE: $\|Tx\|<\|x\|$ for each $x\in H\setminus\{0\}$ $\|T\|\leq1$ and ...