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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).

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, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

52582 questions
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How to define $f(0)$ when $f$ is a function in $L^2$?

Any function $f$ in $L^2$ is a actually an equivalence class and has properties that only hold "almost everywhere." But it would be convenient to speak of the value of $f$ at certain points like $f(0)$. Is there a meaningful way of defining this?
Mark
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About equivalent norms on a vector space

Definition. A norm $\|\cdot\|$ in a vector space $X$ is said to be equivalent to a norm $\|\cdot\|_0$ on $X$ if there are positive numbers $a$ and $b$ such that for all $x \in X$ we have $$ a\| x \|_0 \leq \|x\| \leq b\|x\|_0 $$ My question. If two…
Jonathan G.
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Proving linearity of an operator using boundedness.

I am considering an operator $K\colon \ell^2 \to \ell^2$ given by $$Kx = \sum_{n=1}^\infty e^{-n} \langle x , e_n\rangle e_n $$ where $e_n = (\delta_{k,n})_{k\in \mathrm{N}}$ is the standard basis on the sequence space $\ell^2$ and $ \langle \cdot ,…
Jakob
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The dual of the space of all the bounded functions

I 'd like to know what is the dual space of the space of all the bounded functions on the set $X$, where $X$ can be any set. Also, I don't assume that the function $f$ is measurable relative to any sigma-field. (Thus, underlying sigma-algebra is…
gejikeiji
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To Show Closedness of a Graph in an Application of Closed Graph Theorem

Here's an old exam question I am struggling with: Let E be a Banach space and $ (x_n)_{n \in N} \subset E $ such that $ \sum_{n=1} ^{\infty} | \langle x_n , x^* \rangle | < \infty $ for all continuous linear functionals $ x^* \in E^* $. Show…
kurmee
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Constructing a subset of $\ell_2$ with dense linear span and infinite complement

the problem I'm stuck on is the following: Suppose that S is a countably infinite subset of $\ell_2$ with the property that the linear span of S′ is dense in $\ell_2$ whenever S\S′ is finite. Show that there is some S′ whose linear span is dense in…
S. Dodd
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Application of Banach-Steinhaus theorem

Let $(x_n)$ be a sequence in a Banach space $E$ such that $\sum_{j=1}^{\infty} |\varphi (x_j) |<\infty$, $\forall \varphi \in E'.$ Then $\sup \limits_{\|\varphi\| \leq 1} \sum_{j=1}^{\infty}|\varphi (x_j)| <\infty $. My attempt: For all $n \in…
Santos
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show that the image of the unit ball under $T$ is not compact.

Let $C[0,1]$ denote the Banach space of continuous real functions from $[0,1]\to \mathbb{R}$. Fix a non constant function $g:[0,1] \to [0,1]$. Define $T: C[0,1] \to C[0,1]$ by $[T(f)](x) = f(g(x))$ for $x \in [0,1]$, ie, $T(f) = f \circ g$. Show…
Jaimini
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weak-$*$ topology on $X^{**}$

In Folland, Exercise 5.52(c), the question is to show that the relative topology on $X$ induced by the weak-$*$ topology on $X^{**}$ is the weak topology on $X$. It is not clear to me what is meant by weak-$*$ topology on $X^{**}$. If I understand…
steve
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Show that $l_\infty$ is not separable.

I'm looking at the proof of showing that $l_\infty$ is not separable, but there is a minor detail I don't understand. In the proof below, is it necessary to define $z_n$ as such instead of simply defining $z_n=x_n^n +1$? I don't understand why we…
nomadicmathematician
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Homeomorphism between a normed space and its open unit ball

I have studied that every normed space $X$ is homeomorphic to its open unit ball $B$. I want to know what conclusion can we draw from this statement? Does it mean that every normed space is open ball? I am confused with this statement. Could anyone…
Srijan
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Parseval relation on inner product space for $\langle x,y \rangle$

Exercise 3.6-4 in Kreyszig asks to show that $\langle x,y \rangle = \sum_k \langle x,e_k \rangle \overline{\langle y,e_k \rangle}$ using the "Parseval relation": $\sum_k |\langle x, e_k \rangle |^2 = ||x||^2$, for all $x \in X$, where the $(e_k)$…
Fequish
  • 709
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Compactness of an Integral operator

Let $K(s,t)$ be a real-valued function of two real variables, and let $T: L^2(\mathbb{R}) \to L^2(\mathbb{R})$ be defined by $(Tf)(s) = \int_\mathbb{R} K(s,t) f(t) dt$. If $||K||_{L^2({\mathbb{R}^2})} < \infty$, can we say that $T$ is a compact…
user1736
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Separable version of Banach-Alaoglu

My notes say that the theorem of Banach-Alaoglu states the following: If $X$ is a normed separable space, then every bounded sequence in $X'$ has a weak-* convergent subsequence. How is this equivalent to the usual formulation from Wikipedia, etc -…
amaflix
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Is $f$ injective in $W$?

If $\|f(x)-f(y)\|\geqslant \frac 1{2} \|x-y\|$ for any $x, y \in W$ then $f$ is injective in $W$ How to prove this? If that inequality is right is it mean that the images are equal or not?
Kamil Hismatullin
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