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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Spectrum of the operator

Let $T$ be an operator on Hilbert space. Define $\sigma(T)=\lbrace \lambda\in \mathbb{C} | \lambda I - T~\textrm{is not invertible}\rbrace$. How can I prove that $\sigma(T^n)=\lbrace ...
2
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1answer
266 views

Relation between Sobolev Space $W^{1,\infty}$ and the Lipschitz class

I have a Sobolev space related question. In the book 'Measure theory and fine properties of functions' by Lawrence Evans. I know the result that states that for $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
2
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1answer
1k views

Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
2
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1answer
147 views

Please help me prove: $v(a+b)\leq v(a)+v(b)$, and $v(ab)\leq v(a)v(b)$ where $v(x)=\inf{\{\vert x^n \vert}^{1/n}: n\in\mathbb{N}\}$

I'm reading a book functional analysis, and reading and have seen an example of somebody please help me if you can. The example that I've seen is the following: If $A$ is a normed algebra and ...
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1
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190 views

Show $\{u_n\}$ orthonormal, A compact implies $\|Au_n\| \to 0$

I'm having a bit trouble with this homework exercise. Let $\mathcal{H}$ be a Hilbert space and $\{u_n\}_{n=1}^\infty$ an orthonormal sequence in $\mathcal{H}$. Let $A$ be a compact operator on ...
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1answer
766 views

A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
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2answers
268 views

Representation of a linear functional in vector space

In the book Functional Analysis,Sobolev Spaces and Partial Differential Equations of Haim Brezis we have the following lemma: Lemma. Let $X$ be a vector space and let $\varphi, \varphi_1, \varphi_2, ...
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69 views

Part (a) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (a) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
4
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1answer
67 views

Assume $T$ is compact operator and $S(I- T) = I $.Is this true that $(I- T)S =I$?

Suppose $S,T \in {\rm B}(X)$ and assume $T$ is compact operator and $S(I- T) = I $.Is this true that $(I- T)S =I$?
4
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1answer
925 views

Weak topology on an infinite-dimensional normed vector space is not metrizable

I've been pondering over this problem for a while now, but I can't come up with a proof or even a useful approach... Let $X$ be am infinite-dimensional normed vector space over $\mathbb{K}$ (that is ...
4
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703 views

how to show that the norm-limit of compact operators is compact?

When I read the item of compact operator on Wikipedia, it said that Let $T_{n}, n\in \mathbb{N}$, be a sequence of compact operators from one Banach space to the other, and suppose that $Tn$ ...
4
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2answers
749 views

The Space $C(\Omega,\mathbb{R})$ has a Predual?

Let $\Omega$ be a compact metric space and $C(\Omega,\mathbb{R})$ the space of borel continuous real valued functions. I would like to know if there is any real Banach space $V$ such that its dual ...
4
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1answer
428 views

$\ell^1$ vs. continuous dual of $\ell^{\infty}$ in ZF+AD

Let the base field be the real numbers or the complex numbers (I don't think it will matter). Let $(\ell^{\infty})'$ be the continuous dual of the Banach space $\ell^{\infty}$. Let $\: f : \ell^1 ...
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2answers
562 views

Proof of $f = g \in L^1_{loc}$ if $f$ and $g$ act equally on $C_c^\infty$

Let $f$ and $g$ be locally integrable, say on $R^n$ (for arbitrary open domains, just extend trivially). Suppose $\forall \phi \in C_c^\infty : \int f \phi dx = \int g \phi dx$. Let $K = ...
3
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1answer
491 views

$\omega$ - space of all sequences with Fréchet metric

I'm working on to prove the following: Show that the convergence in the space $\omega$ (space of all sequences with respect to the Fréchet metric) is the coordinate convergence. Any hint is ...
2
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1answer
101 views

Selfadjoint Restrictions of Legendre Operator $-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$

Problem: Let $Lf =-((1-x^{2})f')'$ be the Legendre differential operator defined on the domain $\mathcal{D}(L)$ consisting of twice absolutely continuous functions on $(-1,1)$ for which $f, Lf \in ...
2
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1answer
867 views

A compact operator is completely continuous.

I have a question. If $X$ and $Y$ are Banach spaces, we have to prove that a compact linear operator is completely continuous. A mapping $T \colon X \to Y$ is called completely continuous, if it maps ...
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141 views

Normal Operators: Numerical Range

Disclaimer: As I realized in the comments that this works for normal operators I decided to modify this question. Besides, I got the proof now - thanks to T.A.E.! Prove that for normal operators the ...
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2answers
71 views

show that the element in $\ell^1$.

If $x=(x_n)$ is a sequence of complex numbers such that the series $\sum x_ny_n$ is convergent for all $y=(y_n)\in{c_0}$. Then prove that $x\in{\ell^1}$. Can anyone tell me what is the meaning of ...
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1answer
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Generalized Riemann Integral: Improper Version

Reference For a bounded nonexample of integrability see: Riemann Integral: Bounded Nonexample For a convergence theorem on integral see: Riemann Integral: Uniform Convergence For a comparison of ...
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Integral eigenvectors and eigenvalues

I need to find the eigenvalues e eigenvectors of this integral. a) $$\int_{0}^{2\pi}(\cos^2(x+y)+1/2)\phi (y)dy$$ b)- Solved thanks $$\int_{0}^{1}(x^2y^2-2/45)\phi (y)dy$$
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3answers
990 views

Instructive proofs in functional analysis

I am beginning to learn functional analysis (from Folland and Royden), but I am from a non-mathematical background, so I often encounter techniques in proofs that I am not familiar with (for example ...
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What is the difference between a Hamel basis and a Schauder basis?

Let $V$ be a vector space with infinite dimensions. A Hamel basis for $V$ is an ordered set of linearly independent vectors $\{ v_i \ | \ i \in I\}$ such that any $v \in V$ can be expressed as a ...
17
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A function that is $L^p$ for all $p$ but is not $L^\infty$?

Let $X$ be the interval $[0,1]$ with Lebesgue measure. Is there a function $f\in L^p(X)$ for all $p\in[1,\infty)$ that is not $\in L^\infty(X)$? If so, what is an example? Motivation: In a course ...
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1answer
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Acting of a dual pairing in Sobolev Spaces

I have a question about the Sobolev Space $H^1_0(U)$, where $U$ is a open subset of $\mathbb{R}^n$. Let us denote with $H^{-1}(U)$ the dual space of $H^1_0$. How is the acting of $H^{-1}$ and $H^1_0$ ...
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4answers
495 views

What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
14
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1answer
701 views

When is $L^1 = (L^\infty)^\ast$?

I found this exercise in Cohn's Measure Theory: Let $(X, \mathscr A, \mu)$ be a finite measure space. Show that the conditions the map $T: L^1(X, \mathscr A, \mu) \to (L^\infty(X, \mathscr ...
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1answer
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Eigenvalues of matrix with entries that are continuous functions

For each $t \in [0,b]$, let $M(t)$ be an $n \times n$ matrix with entries $m_{ij}(t).$ The matrix $M(t)$ is invertible and positive-definite, so the eigenvalues of $M(t)$ exist and are positive for ...
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Differentiating an Inner Product

If $(V, \langle \cdot, \cdot \rangle)$ is a finite-dimensional inner product space and $f,g : \mathbb{R} \longrightarrow V$ are differentiable functions, a straightforward calculation with components ...
12
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When is the image of a linear operator closed?

Let $X$, $Y$ be Banach spaces. Let $T:X\rightarrow Y$ be a bounded linear operator. Under what circumstances is the image of $T$ closed in $Y$ (except finite-dimensional image). In particular, I ...
23
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1answer
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On the norm of a quotient of a Banach space.

Let $E$ be a Banach space and $F$ a closed subspace. It is well known that the quotient space $E/F$ is also a Banach space with respect to the norm $$ \left\Vert x+F\right\Vert_{E/F}=\inf\{\left\Vert ...
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Dual of a dual cone

Any hint on how to prove the following please: Let $K$ be a convex cone, and $K^*$ its dual cone. Prove that $K^{**}$ is the closure of $K$. Thanks!
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A theorem due to Gelfand and Kolmogorov

For any topological space $X$, we can define $C(X)$ to be the commutative ring of continuous functions $f\,:\,X\rightarrow \mathbb{R}$ under pointwise addition and multiplication. Then $C(-)$ becomes ...
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4answers
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Is the space $C[0,1]$ complete?

In order to prove $C[0,1]$ is complete, my functional analysis book says: "It is only necessary to show that every Cauchy sequence in $C[0,1]$ has a limit". It goes on by supposing $\{x_n\}$ is a ...
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Example of different topologies with same convergent sequences

It's well known that for metric spaces the following is true Let $ X $ be a space with two different metrics $ d_1,d_2$ such that the two topological spaces $ (X,d_1),(X,d_2) $ have the same ...
11
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2answers
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$T$ is continuous if and only if $\ker T$ is closed

Let $X,Y$ be normed linear spaces. Let $T: X\to Y$ be linear. If $X$ is finite dimensional, show that $T$ is continuous. If $Y$ is finite dimensional, show that $T$ is continuous if and only if $\ker ...
10
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1answer
428 views

Selfadjoint compact operator with finite trace

I have a compact selfadjoint operator $T$ on a separable Hilbert space. For some fixed orthonormal basis, the operator's diagonal is in $\ell^1(\mathbb{N})$. Can we conclude that $T$ is trace ...
9
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1answer
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Why is $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$?

In Lieb and Loss's Analysis, I saw that they mentioned $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$ (dense wrt the $L^2$ norm, I think). But I didn't find its proof in the ...
8
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1answer
464 views

reference for operator algebra

I am taking a course on operator algebra this semester. My instructor has suggested a reference "Kadinson and Ringrose." Are there any other good/standard references for this subject that I can look ...
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539 views

Is compactness a stronger form of continuity?

Let $H$ be a Hilbert space. We say that a linear operator $T \colon H \to H$ is compact if it maps bounded sets to precompact ones, that is, if for every bounded sequence $(a_n)$ in $H$, $(Ta_n)$ has ...
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spectrum of the right shift operator

Here is the question: Considering the right shift operator $S$ on $l^2({\bf Z})$, what can one know about ran$(S-\lambda)$? Here is what I thought: If one wants to prove that the operator ...
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731 views

Must a weakly or weak-* convergent net be eventually bounded?

Let $\mathfrak{X}$ be a Banach space. As a standard corollary of the Principle of Uniform Boundedness, any weak-* convergent sequence in $\mathfrak{X}^*$ must be (norm) bounded. A weak-* convergent ...
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Why isometric isomorphic between Banach spaces means we can identify them?

Is the "isometric" part really necessary? For what reason is that? Eg. we prove that there is an isometric isomorphism between $(L^p)'$ and $L^q$ ($(p,q)$ conjugate) and then we identify them ...
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Example of a closed subspace of a Banach space which is not complemented?

In this post, all vector spaces are assumed to be real or complex. Let $(X, ||\cdot||)$ be a Banach space, $Y \subset X$ a closed subspace. $Y$ is called $\underline{\mathrm{complemented}}$, if there ...
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1answer
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Properties of dual spaces of sequence spaces

Can you tell me if I got the following homework right? Nitpicking is welcome. a) Recall that $$ c_0 (\mathbb{N}) = \{ f: \mathbb{N} \rightarrow \mathbb{C} \mid \lim_{n \rightarrow \infty } f(n) ...
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1answer
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Prove: The weak closure of the unit sphere is the unit ball.

I want to prove that in an infinite dimensional normed space $X$, the weak closure of the unit sphere $S=\{ x\in X : \| x \| = 1 \}$ is the unit ball $B=\{ x\in X : \| x \| \leq 1 \}$. $\\$ Here is ...
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What is the topological dual of a dual space with the weak* topology?

I'm trying to understand a claim I heard in class. To be concrete, suppose $X$ is a compact, hausdorff topological space, and let $C(X)$ be the space of continuous functions on $X$ with the supremum ...
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Proving $\ell^p$ is complete

Let be $1\leq p\in\mathbb{R}$, denote: $$\ell^p(\mathbb {R})=\left\{(x_n)\subset \mathbb{R}: (x_n) \mbox{ is a sequence with } \displaystyle\sum_{n=1}^{\infty}|x_n|^p<\infty \right\}$$ ...
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2answers
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$\operatorname{Range}T$ is a closed subspace.

Let $X,Y$ two Banach spaces. If $T \in \mathcal{B}(X,Y)$ study if $\operatorname{Range}T$ is a closed subspaces. How can I prove this fact ? What theorems can I use ? thanks :)