# Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

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### Is the set $V=U\cap-U$ balanced?

Let $E$ be a topological vector space and $U$ be an arbitrary neighborhood of $0$. I would like to know if $V=U \cap -U$ is balanced, that is $\lambda V \subset V$ for all $\lambda \in \mathbb{C}$ ...
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### Need help understanding supremum notation in a distance metic on the set of bounded complex sequences.

I'm working on an excersize (from Kreyszig's Functional Analysis book) trying to prove the triangle inequality for a particular distance metric on the set of bounded complex sequences. I think my ...
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### Is H.L. Royden's REAL ANALYSIS, 4th edition, suitable for these two introductory functional analysis courses?

Is the book Real Analysis by H.L. Royden, 4th edition, suitable for two introductory functional analysis courses comprising the following topics? First Course: Metric spaces: A quick review, ...
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### Let $X= L^2([0,1],\#)$ ( where $\#$ is the measure that counts) $g\in L^\infty([0,1],\#)$ $A(f)(x)=f(x)g(x)$. Calculate spectrum, eigen values..

Let $X= L^2([0,1],\#)$ ( where $\#$ is the measure that counts) $g\in L^\infty([0,1],\#)$ and $A \in \mathcal{L}$ as $A(f)(x)=f(x)g(x)$. i)Show $\sigma(A) = \overline{g([0,1])}$ ii) Determine the ...
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### Zeros of linear combination of basis functions with infinite zeros

Suppose to consider a linear combination $f$ of real functions which are known to have infinitely many zeros on the real line (namely, I am considering the prolate spheroidal wave functions). What can ...
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### Weak-star topology on probability measures

I'm currently starting to study Robust Statistics (based on Huber's book) and still somewhat struggling with the notion of weak-star topology $\tau_\ast$ on the space of probability measures. Huber ...
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### Is this the Gateaux differential of $F(u, v)$?

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and suppose $X(\Omega) =X_1(\Omega)\times X_2(\Omega)$ be a Banach space. Moreover let $p, q\geq 1$ be two real numbers. Suppose the functional ...
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### Powers of (general) closed Operators closed?

I have the following question. Let $X$ be a Banach space (you may specify further properties such as reflexivity or Hilbert space structure if needed) and let $A: \mathcal{D}(A) \to X$ be a closed ...
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### Compactness without using Heine-Borel in $L^p$ spaces

Consider the set of functions $S=\{\sin(2^nx):n\in\mathbb{N}\}$ in $L^2[-\pi,\pi]$ with the metric $d(f,g)=\left(\int_{\pi}^{\pi}|f(x)-g(x)|^2dx\right)^{\frac1{2}}$. Then is $S$ both closed and ...
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### Properties about topological vector spaces

Let $E$ be a topological vector space. First I want to prove that, given a $V \subset E$ balanced and $\lambda>0$ then $$\lambda V \subset \beta V, \: \forall \;\lambda< \beta. \tag{1}.$$ For ...
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### Spectrum of an element in the disk algebra.

Consider the disk algebra $A$ of continuous function on the unit disk $D$ that are analytic on the interior of the disk. Is it true that $\sigma_A(f) = f(D)$ for $f \in A$? A simple yes or no suffices....
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### Can I show this using a contraction semigroup property?

Let $H$ be a (real) Hilbert space, $L$ be an unbounded operator on $H$ with its domain $D(L)$ and $(e^{-tL})_{t\ge 0}$ be a contraction semigroup on $H$. Then, the following holds from a semigroup ...
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### If $0 \in \sigma_c(A) \bigcup \sigma_r(A)$ then is the linear map $A$ is not open

i) Let $X$ be a Banach space and $A \in \mathcal{L}(X)$ such that $0 \in \sigma(A)$ Show that if $0 \in \sigma_c(A) \bigcup \sigma_r(A)$ then is the linear map $A$ is not open. ii) Examples of ...
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### extension of integral preserving positive operators on $L^p$ into $L^q$

Let $(\Omega,\mu)$ be a finite measure space. Let $T \colon L^\infty(\Omega) \to L^\infty(\Omega)$ be a weak* continuous contractive positive operator such that $\int_\Omega T(f)=\int_\Omega f$ for ...
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### continuity of pointwise limit of continuous functions

Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of continuous functions from $[0,1]$ to $\mathbb{R}$, and $\{f_n\}_{n=1}^{\infty}$ converges pointwise to $f$ i.e. $\lim_{n \to \infty} f_n(x) =f(x)$. Define ...
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### How can we characterize weak convergence in $(c, \Vert \, \Vert _{\infty})$?

Let me recall $c = \{ (x_h)_{h \in \mathbb{N}} \subset \mathbb{R} \, | \, \lim_{h \to \infty} x_h = k < \infty \}$ the space of convergent sequences equipped with $\Vert \, \Vert_{\infty}$. What ...
### Let $X$ be a banach space, and let $U$ be a finite dimensional subspace, then there is a closed subspace $V$ s.t $X=U\bigoplus V$
Let $X$ be a banach space, and let $U$ be a finite dimensional subsapce, then there is a closed subspace $V$ s.t $X=U\bigoplus V$ MY attempt: Let $U=Span\{v_1,...,v_n\}$ and consider the following ...