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 Operators: Decomposition

This is a real question of me. Given a Banach space. Consider a basis on finite dimensional range: $$\dim\mathcal{R}F<\infty:\quad y_1,\ldots, y_N$$ Hahn-Banach lifts the dual basis up: $$ y_n\in ...
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1answer
11 views

Why are functional analysts interested in not only the point spectrum of $f$, but also, its spectrum?

Suppose $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\},$ that $X$ is a Banach space over $\mathbb{K}$, and that $f : X \leftarrow X$ is a bounded linear transform. Then the spectrum of $f$ is defined as the ...
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2answers
13 views

Understanding part of a theorem of Calculus of Variations

I have trouble understanding the following statement (From Gelfland's Calculus of Variations book): If $\phi[h]$ is a linear functional and if ...
2
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1answer
17 views

Proof Involving Generalized Mean

Let $x=(x_1,...,x_n) \in \mathbb R^n$ and $$g(p)=\sqrt[p]{\frac{1}{n}\sum_{k=1}^{n} |x_k|^p)}$$ Using Hölder's inequality, show that $g(p)$ is increasing on $(0,\infty)$. For a sequence with ...
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1answer
20 views

Proving a metric on X.

Let $X$ be the collection of all sequences of positive integers. If $x=(n_j)_{j=1}^\infty$ and $y=(m_j)_{j=1}^\infty$ are two elements of $X$, set $$k(x,y)=\inf\{j:n_j\neq m_j\}$$ and $$d(x,y)= ...
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21 views

For a Banach algebra $\mathcal{A}$ and an idempotent $e$, show that $\mathcal{A}e$ is a Banach algebra

I am studying the spectral theory of operators on Banach and Hilbert spaces, reading through Conway's A Course in Functional Analysis. In section VII.4, Conway states Proposition 4.11 as a consequence ...
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15 views

Is the system $\left\{\frac{e^{i n t}}{2\pi}\right\}_{n\in\mathbb Z}$ a Riesz basis on $L^2(-\pi,\pi)$?

Is the system $$\left\{\frac{e^{i n t}}{2\pi}\right\}_{n\in\mathbb Z}$$ a Riesz basis on $L^2(-\pi,\pi)$? I think not because $$\frac{1}{2\pi}\int_{-\pi}^\pi \frac{e^{i (n-m) t}}{4\pi^2}dt\neq 1$$ if ...
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19 views

Why isn't the completion of $C^0$ wrt. the $L^2$ norm a space of sequences instead of a space of functions?

We know that $L^2(\Omega)$ can be defined as the completion of $C^0(\Omega)$ with respect to the norm $$\left(\int_\Omega |u|^2\right)^{\frac 12}.$$ But strictly speaking, $L^2(\Omega)$ is a space of ...
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27 views

Fourier transform of $e^{-\frac{x^2}{2}}\frac{1}{\mathsf{sinc}( x )} $

I have to find inverse Fourier transform of \begin{align*} F(\omega)=e^{-\frac{\omega^2}{2}}\frac{1}{\mathsf{sinc}( \omega )} \end{align*} I was wondering whether it exists maybe in a sense of ...
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34 views

Verifying a Vector Space Via Given Axioms

Let $X$ be the collection of all sequences $\{\alpha_n\}_{n=1}^{\infty}$ of scalars from $\mathbb{K}$ such that $\alpha_n=0$ for all but a finite number of values of $n$. Define addition and scalar ...
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1answer
12 views

Question about a proof of Riemann localization theorem

The Riemann Localization Theorem states that Let $f \in L_{2 \pi}^2$ and $x_0 \in \mathbb R$. Then $$ \lim_{n \to \infty} (S_nf)(x_0) = f(x_0)$$ if and only if there is a $\delta \in (0, \pi)$ ...
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1answer
23 views

Why is $J(u) := \int_\Omega |\nabla u|^2$ convex?

Define $J(u) := \int_\Omega |\nabla u|^2$ over $\{ u \in H^1(\Omega) : tr(u) = g\}$. Why is $J$ convex? I keep getting $J(tu + (1-t)v) \leq 2t^2J(u) + 2(1-t)^2J(v)$ by using the triangle inequality ...
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1answer
25 views

Understanding a proof from Conway: showing existence of idempotents using functional calculus

I am studying the spectral theory of operators on Banach and Hilbert spaces, making use of Conway's A Course in Functional Analysis. In section VII.4, Conway states Proposition 4.11 as a consequence ...
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1answer
24 views

$T$ is bijective and homeomorphism.

Suppose $X$ be the set of all polynomial with real coefficients in one variable with norm $$\|p(x)\|=|a_0|+|a_1|+\dots+|a_n|$$ where $p(x)=a_0+a_1x+\dots+a_nx^n$ which induces a metric ...
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1answer
16 views

which of the following sequences $\{f_n\}\in C[0,1]$ must contain a uniformly convergent subsequence?

Could anyone tell me which of the following sequences $\{f_n\}\in C[0,1]$ must contain a uniformly convergent subsequence? $|f_n(t)|\le 3\forall t\in [0,1],\forall n$ $f_n\in C^1[0,1],|f_n(t)|\le ...
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19 views

Is this a Cauchy sequence?

Let $Y$ and $Z$ be Banach spaces. Define $|u|_X := |Au|_Y + |Bu|_Z$ where $A$ and $B$ are linear maps. Suppose I have a sequence $(u_n)$ such that $|Au_n|_Y \to 0$ and $|Bu_n - Bu_m|_Z \to 0$. Does ...
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0answers
14 views

How to prove this sequence is in $l^2$? [duplicate]

I ran into such a problem in some exercise book on hilbert space. Suppose we have a sequence $ \{a_n \}_1^\infty$. Now, for any sequence $\{ b_n \}_1^\infty $ in $l^2$, we have $$ ...
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1answer
12 views

Short proof of sequential Banach Alaoglu for Hilbert spaces

Do you know of a short proof of the fact that bounded sequences in Hilbert spaces admit weakly converging subsequences? If the space is separable, then the common sequential-version proof is what I ...
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1answer
23 views

On the greatest norm element of weakly compact set

Let $X$ be a Banach space and $K\subset X$ be a nonempty weakly compact set. I would like to know if there exists a point $u_0\in K$ such that $\|u_0\|\geq \|u\|$ for all $u\in K$. Thank you for all ...
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2answers
73 views

Proving a subset of $l_2$ is closed

Let $l_2$ be the set of all real sequences $x=(x_n)$ such that $\sum|{x_n}|^2 <\infty$ and define the norm $||x_n||_2=(\sum\limits_{n=1}^{\infty}|x_n|)^{\frac{1}{2}}$. I want to show that $A=\{ ...
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14 views

problem about equivalent norms. [duplicate]

Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be equivalent norms on a normed field. Then (i) $\|x\|_1<1$ iff $\|x\|_2<1$; $\|x\|_1>1$ iff $\|x\|_2>1;$ (ii) $\|x\|=1$ iff $\|x\|_2=1$. I want to ...
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1answer
23 views

Embedding $L^2[0,1]$ into any Hilbert space?

Is it true that every Hilbert space has a closed subspace isometrically isomorphic to $L^2[0,1]$? Can someone sketch a proof of this, or at least point me in the right direction to understanding it? ...
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1answer
48 views

Approximation Property: Hilbert Spaces [on hold]

Note: This thread is not to gain reputation!! Given a Hilbert space. How to prove: It has the approximation property!
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18 views

Approximation Property: Characterization

Problem Given a Banach space. Suppose it has the approximation property: $$C\in\mathcal{C}:\quad\|T_\varepsilon-1\|_C<\varepsilon\quad(T_\varepsilon\in\mathcal{F}(E))$$ Then every compact ...
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1answer
27 views

about a sequence of isometries' convergency.

Let $M$ be a compact metric space, let $(i_n)$ be a sequence of isometries: $M \rightarrow M$. I've already showed that there exists a subsequence $(i_{n_k})$ that converges to $i$ which is also a ...
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0answers
12 views

When is the completion of a topological vector space a Frechet space?

Suppose $X$ is a topological vector space with the metric topology. If we take the completion of $X$ with respect to the metric, will we get a Frechet space? Are there any extra conditions needed to ...
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33 views

Prove that any function f in $L^p$ is the limit, in the metric of $L^p$, of a sequence of simple functions.

I know that I'll need to use dominated convergence here. In the problem, they ask to consider, when f is bounded and nonnegative, the sequence: $s_n(x) = \begin{cases} \frac{i-1}{2^n} \text{ for } ...
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1answer
23 views

If $f_n \rightarrow f$ in $L^p$ and $g_n \rightarrow g$ in $L^q$, where $\frac{1}{p} + \frac{1}{q} = 1$, show that $f_n g_n \rightarrow fg$ in $L^1$ [duplicate]

I know that this will have something to do with Holder's inequality but I am at a loss as to how the $L^p$ and $L^q$ convergence in $f$ and $g$ dictate the convergence in $L^1$. Any help is ...
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1answer
32 views

Inverse operator of $I-A$

Let $H$ be an Hilbert space, $A:H\to H$ be a bounded linear operator such that $$ \|A^{n_0}\|< 1\qquad\text{for some}\quad\; n_0\in\mathbb{N}. $$ I have to show that $I-A$ is invertible. My idea ...
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22 views

Check Functional Analysis Proof

I seem to have proved something with elementary techniques even though the paper I found it in suggests it requires heavy tools. There could be a mistake but I can't find it if there is one. Theorem: ...
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1answer
21 views

weak -star topology on ball M[0,1] [on hold]

let B = ball M[0,1] and for $ \mu$, $v$ in M[O,1] define $ d( \mu, v) = \sum_{n=0} ^{\infty} 2^{-n} \vert \int _0 ^1 x^n d \mu - \int _0 ^1 x^n d v \vert $ .Show that d is a metric on M [0, 1] that ...
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1answer
15 views

Partial Isometries: Subspaces

Note: This thread is not to gain reputation!!! Given an operator algebra. Then a partial isometry satisfies both: ...
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2answers
59 views

Series convergence and compact space

Let $K$ be a compact topological Hausdorff space. $\{x_n\}_1^\infty \subset K $ such that $x_i \not= x_j, i \not=j$ and $\{a_i\}_1^\infty \subset \mathbb{K}$. Show the folowing are equivalent: for ...
2
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1answer
20 views

Adjoints of operators between different Hilbert spaces.

When we have an operator $$ T ~\colon \mathscr{H} \longrightarrow \mathscr{H} $$ from a Hilbert space to itself, we can use the Riesz representation theorem to prove the existence of the adjoint ...
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1answer
20 views

Partial Isometries: Characterization

Note: This thread is not to gain reputation!!! Given a C*-algebra. Any partial isometry satisfies: $$WW^*W=W$$ From this, one derives projections: $$W^*W,WW^*$$ Conversely, given projections: ...
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15 views

$C(K)^*$ when K is a countable, compact metric space. [on hold]

If K is a countable, compact metric space, then why $C(K)^*$ consists of only purely atomic measures? Also, why $C(K)^*$ is isometric to $\ell_1$ ? (See Topics in Banach space theory by Albiac and ...
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7 views

Separating and cyclic vector

Let $\{\Gamma_i , \mu_i\}_{i\in I}$ be a family of probability measure spaces and suppose $I$ is uncountable. Let $\{\Gamma , \mu\} = \prod_{i\in I} \{\Gamma_i,\mu_i\}$ be the product measure space. ...
3
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2answers
39 views

Logic behind a proof in Topological Vector Spaces

I found the following result at the beginning of some notes on topological vector spaces (TVS). This is a quite fundamental result, that apparently is considered the corresponding version of the ...
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1answer
18 views

Semi-norm on essentially bounded functions

Consider the space of essentially bounded functions (before quotienting it to create the $L^\infty$ space). On that space, I read, $|||.|||_\infty$ is only a semi-norm. So I wanted to find an example, ...
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1answer
19 views

Direct sum and intersection of sets [on hold]

If $A_1,A_2,B_1,B_2$ are subspaces of a Hilbert space $H$, is the following statement true or not? $$\left( A_1 \oplus A_2\right)\cap \left( B_1 \oplus B_2\right)=\left(A_1\cap B_1\right)\oplus ...
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0answers
24 views

Finding roots of a discrete complex valued function [on hold]

I am struggling with a numerical problem. I have a discrete dataset with complex valued numbers which are the function of a real variable. The function is a black box. Is there any way to find the ...
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0answers
13 views

Irreducible representation and rank one projetion

Let $A$ be a C*-algebra with a nonzero minimal projection $e$. a - Show that if $\{\pi, H\}$ is an irreducible representation of $A$ such that $\pi(e) \neq 0$, then $\pi(e)$ is a projection of rank ...
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1answer
15 views

A symmetric algebra that is not a C* algebra

Recall that a commutative Banach $*$-algebra $A$ is called symmetric if the Gelfand transform replaces involution in $A$ by complex conjugation in $\mathbb{C}$. Moreover, any commutative C* algebra is ...
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13 views

Prove that $C_c(X)^c=C_0(X)$ if $X$ is locally compact $T_2$ [on hold]

Prove that $C_c(X)^c=C_0(X)$ if $X$ is locally compact $T_2$ So in genarel $C_c(X)$ is not complete wr.t . Supnorm.
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1answer
26 views

Prove that the space of sequences with limit $0$ is complete.

Prove that $C_0$ (the space of sequences with limit $0$) is complete. My effort: Let {$x_n$} be sequnce in $C_0$ converging to the limit $0$. As the {$0$} is in $C_0$ hence $C_0$ closed in $C$ ...
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0answers
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Gateaux but not Frechet differentiable functional [duplicate]

For functional between Banach spaces X,Y: By Gateaux differentiable at $u\in X$ I mean that there exists bounded linear operator $dF(u)$ s.t. $F(u+t\xi)-F(u)=dF(u)\xi+o(t)$ for all $\xi\in X$. For ...
3
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2answers
44 views

Is this a bounded linear map?

I tried very hard to (dis)prove it, but now I give up. Define a map which maps $x\in L_2[0,1]$ to the function $$(Tx)(t) = \frac{1}{\sqrt{t}}\int_0^t \frac{x(s)}{\sqrt{s}} \,d s.$$ I don't even ...
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1answer
18 views

How to compute the $H^{-s}(\Omega)$-norm of a function?

Suppose to have a sufficiently regular domain $\Omega\subseteq\mathbb{R}^d$. I know that, for $s\in\mathbb{R}_+$, the space $H^{-s}(\Omega)$ is defined as the dual of $H^s_0(\Omega)$, endowed with the ...
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0answers
32 views

$-\Delta u = f$ in $L^2(0,T;H^{-1}(\Omega))$ (as opposed to $H^{-1}(\Omega)$)

Why does nobody consider the equation $-\Delta u = f$ in the space $L^2(0,T;H^{-1}(\Omega))$? Eg. given $f \in L^2(0,T;L^2(\Omega))$ find a solution $u \in L^2(0,T;H^1_0(\Omega))$ such that ...
2
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0answers
35 views

Is this a Hilbert space? If not, is it reflexive?

Let $E$ be a Banach space. Let $L^2(\Omega, E)$ denote the space of random variables taking values in $E$ with second order moment. Is $L^2(\Omega,E)$ a Hilbert space? or at least, reflexive? 1) I do ...