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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Equivalent Definitions of the Operator Norm

Would you give me a proof of the equivalence of these ones? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\ ...
12
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1answer
2k views

Weak-to-weak continuous operator which is not norm-continuous

Can one give a "relatively easy" example of a linear mapping $T\colon X\to X$ ($X$ a Banach space) which is a) weak-to-weak continuous b) weak*-to-weak* continuous ($X=Y^*$) but not norm-to-norm ...
9
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2answers
829 views

Weak-* sequential compactness and separability

Let $X$ be a Banach space, and let $B$ be the closed unit ball of $X^*$, equipped with the weak-* topology. Alaoglu's theorem says that $B$ is compact. If $X$ is separable, then $B$ is metrizable, ...
9
votes
1answer
588 views

Inclusion of $L^p$ spaces

Let $X \subset L^1(\mathbb{R})$ a closed linear subspace satisfying \begin{align} X\subset \bigcup_{p>1} L^p(\mathbb{R})\end{align} Show that $X\subset L^{p_0}(\mathbb{R})$ for some ...
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2answers
244 views

Can all real/complex vector spaces be equipped with a Hilbert space structure?

Let $X$ be a vector space over $\mathbb K \in \{\mathbb R, \mathbb C\}$. Does there exists a pairing $X \times X \rightarrow \mathbb K$ that induces a Hilbert space structure on $X$? I have been ...
5
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2answers
2k views

How to prove that if a normed space has Schauder basis, then it is separable? What about the converse?

Can we take as a dense subset the collection of all the linear combinations of the vectors of the Schauder basis using the rationals as scalars (or the complex numbers with rational real and imaginary ...
4
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2answers
84 views

$\int_{\mathbb{R}}f(x)e^{-ixz}d\mu_x$ analytic for $f\in L_1$

Let $f\in L_1(-\infty,\infty)$ be a Lebesgue-summable function on $\mathbb{R}$ and let $x\mapsto e^{\delta|x|}f(x)$ also be Lebesgue-summable on all the real line. I have added the condition that ...
11
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4answers
831 views

Is there a continuous bijection between an interval and a square: $[0,1] \mapsto [0,1] \times [0,1]$?

Is there a continuous bijection from $[0,1]$ onto $[0,1] \times [0,1]$? That is with $I=[0,1]$ and $S=[0,1] \times [0,1]$, is there a continuous bijection $$ f: I \to S? $$ I know there is a ...
8
votes
3answers
308 views

For which $s\in\mathbb R$, is $H^s(\mathbb T)$ a Banach algebra?

According to Theorem 4.39 in Adams & Fournier Sobolev Spaces: If $mp>n$, $m\in\mathbb N$, then $W^{m,p}(\Omega)$ is a Banach algebra, provided that $\Omega\subset\mathbb R^n$ satisfies the ...
7
votes
1answer
255 views

Fredholm Equations

I have the following problem to solve $$\phi (x)=\lambda \int_{o}^{\pi }\cos(2x+y)\phi (y) dy+ \sin x$$ following the instructions from the following link early to conclude that: $$\phi (x)=\lambda ...
7
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1answer
473 views

The $L^1$ convergence of $f(x - a_n) \to f(x)$

I'm trying to solve something, and I'm stuck. Suppose you have a function $$f \in L^1(\mathbf R) $$ and a sequence of real numbers that converges to zero: $$ a_n \rightarrow 0 $$ define a sequence ...
6
votes
1answer
576 views

non-continuous function satisfies $f(x+y)=f(x)+f(y)$

As mentioned in this link, it shows that For any $f$ on the real line $\mathbb{R}^1$, $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable. But how to show there exists such ...
6
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2answers
424 views

How to prove that $C^k(\Omega)$ is not complete

Let $\Omega \subset\mathbb{R}^n$ be some bounded domain. And Consider the set of all k-times differentiable functions $C^k(\Omega)$. I want to prove that this set is not complete with the inner ...
4
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1answer
755 views

For positive invertible operators $C\leq T$ on a Hilbert space, does it follow that $T^{-1}\leq C^{-1}$?

I need the following result. I think it's quite obvious but I don't know how to prove that: Let $C, T : \mathcal{H} \rightarrow \mathcal{H}$ be two positive, bounded, self-adjoint, invertible ...
4
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0answers
176 views

An explicit functional in $(l^\infty)^*$ not induced by an element of $l^1$? [duplicate]

Possible Duplicate: Nonnegative linear functionals over $l^\infty$ Setup: Let $l^\infty$ be the set of bounded sequences (with terms in $\mathbb{R}$), and let $l^1$ be the set of sequences ...
3
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3answers
889 views

Space of bounded functions is reflexive if the domain is finite

Let $C_b(X)$ be a space of bounded continuous functions on a locally compact space $X$ equipped with the supremum norm. How to show that $C_b(X)$ is reflexive if and only if $X$ is finite?
2
votes
1answer
117 views

Banach spaces: Convergence in terms of the Schauder basis.

Let $X$ be a Banach space. Suppose $X$ has a normalized Schauder basis $\{x_n\}_{n \in \Bbb N}$. Let $\{y_n\}_{n \in \Bbb N}$ be a sequence in $X$ converging to $0_X$. For each $n \in \Bbb N$, let ...
2
votes
1answer
159 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 ...
2
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2answers
172 views
2
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1answer
692 views

The vector space formed on C[0,1] and the norm $(\int_{0}^{1}|f(t)|^2 dt)^{1/2} $

How does one show that $(C[0,1], ||.||_{2})$ where $||.||_2=(\int_{0}^{1}|f(t)|^2dt)^{1/2}$ and $C[0,1]$ is the space of all continous function which are mapped from $[0,1]\rightarrow \mathbb{R}$, is ...
1
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1answer
86 views

Summary: Spectrum vs. Numerical Range

This thread is only Q&A! Given a Hilbert space $\mathcal{H}$. Consider operators: $$T:\mathcal{D}(T)\to\mathcal{H}$$ Denote for shorthand: ...
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1answer
195 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 ...
7
votes
2answers
286 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, ...
6
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1answer
586 views

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $. Where $A,\ B$ are bounded operators in Banach space and $\sigma$ denotes spectrum.
6
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2answers
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On the weak closure

Let $\lbrace e_n \rbrace$ for the standard unit vectors in $l_2$. I want to show that $0$ is in weak closure of $\lbrace\sqrt{n}e_n\rbrace$ but no subsequence of $\lbrace \sqrt{n}e_n\rbrace$ weakly ...
5
votes
2answers
2k views

Why is $l^\infty$ not separable?

My functional analysis textbook says "The metric space $l^\infty$ is not separable." The metric defined between two sequences $\{a_1,a_2,a_3\dots\}$ and $\{b_1,b_2,b_3,\dots\}$ is ...
5
votes
1answer
2k views

Is any closed ball non-compact in infinite dimensional space?

It is known that the closed unit ball $\overline{B_1(0)}$ in a normed space $X$ is compact if and only if $\dim X < \infty$. In particular, the $\overline{B_1(0)}$ is not compact if $\dim X = ...
4
votes
2answers
503 views

Fubini's theorem and $\sigma$-finiteness?

I'm reviewing my analysis notes, and I am really confused about what is meant by $\sigma$-finiteness being a hidden hypothesis of Fubini's theorem. Here is Fubini's theorem as was stated to me: ...
4
votes
1answer
806 views

Show reflexive normed vector space is a Banach space

$X$ is a normed vector space. Assume $X$ is reflexive, then $X$ must be a Banach space. I guess we only need to show any Cauchy sequence is convergent in $X$.
4
votes
2answers
832 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
490 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 ...
3
votes
1answer
432 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 $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
3
votes
2answers
323 views

Contraction and Fixed Point [duplicate]

How do I show that for $T: X \rightarrow X$ where X is complete and $T^m$ is a contraction that T has a unique fixed point $x_0 \in X$. I know there exists $\lambda_1 \in (0,1)$ for $x, y \in X$ ...
3
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1answer
693 views

If $X^\ast $ is separable $\Longrightarrow$ $S_{X^\ast}$ is also separable

Let $X$ be a Banach space such that $X$* (Dual space of $X$) is separable How can we prove that $S_{X^\ast}$ (Unit sphere of $X$*) is also separable Any hints would be appreciated.
3
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1answer
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When are two norms equivalent on a Banach space?

I'm working on an exercise from functional analysis. Let $E$ be a vector space and $\|\cdot\|_1$ and $\|\cdot\|_2$ be two complete norms on $E$. Now suppose that $E$ satisfies the following property: ...
3
votes
1answer
673 views

For Banach space there is a compact topological space so that the Banach space is isometrically isomorphic with a closed subspace of $C(X)$.

I want to prove that for Banach space V there is a compact topological space $X$ so that $V$ is isometrically isomorphic to a closed subspace of $C(X)$-continuous function on a (compact) topological ...
3
votes
2answers
629 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
votes
1answer
451 views

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
votes
1answer
108 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 ...
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1answer
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Any two norms equivalent on a finite dimensional norm linear space.

I am trying to understand the proof that every two norms on a finite dimensional NLS are equivalent. I am working with this proof I found on the web: ...
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2answers
174 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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1answer
313 views

Complementability of von Neumann algebras

Is every von Neumann algebra complemented in its bidual? It is certainly true for commutative von Neumann algebras as their spectrum is hyperstonian. Is it 1-complemented?
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1answer
58 views

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 ...
5
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1answer
93 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 ...
5
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1answer
81 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 $(I- T)S =I$?
5
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1answer
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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 ...
5
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2answers
847 views

How to show that the limit of compact operators in the operator norm topology 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 $Tn$ ...
3
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1answer
562 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
121 views

If space of bounded operators L(V,W) is Banach, V nonzero, then W is Banach (note direction of implication)

Let $V,W$ be normed vector spaces, and $L(V,W)$ be the space of bounded linear operators. Usually I would only see the statement "If $W$ is Banach, then $L(V,W)$ is Banach.". But Wikipedia writes that ...
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2answers
74 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 ...