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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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A locally convex space is metrizable if and only if its topology is determined by a countable set of seminorms.

A locally convex space is metrizable if and only if its topology is determined by a countable set of seminorms. In the proof of Conway's book, I have a trouble in understanding => direction: Assume that $X$ is metrizable with metric $\rho$. Let…
Gobi
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Bounded Linear functional on $\mathcal{L}_{2}[a,b]$

Lets say that $f:[a,b] \rightarrow \mathbb{R}$ is a measurable function such that $H: \mathcal{L}_{2}[a,b] \rightarrow \mathbb{R}$ defined as $H(g) = \int_{a}^{b}fg$ is finite for all $g \in \mathcal{L}_{2}[a,b]$ I was wondering if $H$ is a bounded…
Mykie
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Show that A has an unique extension to a bounded operator on H

if $\{e_1, e_2, \cdots \}$ is an orthonormal basis for Hilbert space $H$ and for each $n$ there is a vector $Ae_n$ in $H$ such that \begin{equation*} \sum ||Ae_n||<\infty . \end{equation*} Show that $A$ has an unique extension to a bounded…
user29272
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Dominated positive operator

I want that if $H$ Hilbert space where $A$, $B$ are positive operators on $H$ Hilbert space, $0 \leq (Ax|x) \leq (Bx | x)$ $\forall x$, does this mean $(A^2x|x) \leq (B^2x|x)$? Thank you
dominiquebartone1
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Lp Spaces, Functional Analysis

Assume $1
user24367
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Prove that range of operator is closed.

$X$ and $Y$ are Banach spaces and $T$ is a bounded linear operator from $X$ to $Y$ which sends bounded closed sets to closed sets. Prove that $T(X)$ is closed. Here I tried to used the fact that a subspace of a complete metric space is closed if…
Prithviraj Chowdhury
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Usefulness of Functional analysis

I heard that functional analysis can be applied to many problems in signal processing. I'm trying to explain to my engineer friend why it is useful, but I learnt it in a pure math setting. Can anyone give me some insight on how functional analysis…
user119264
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Show that $\operatorname{ran}(I-T)$ is dense in $\ell^2$, $T$ is a right shift operator

$T$ is a right shift operator from $\ell^2 \to \ell^2$, $(\alpha_1, \alpha_2,\ldots)\mapsto (0,\alpha_1,\alpha_2,\ldots)$. I want to show that $\operatorname{ran}(I-T)$ is dense in $\ell^2$. Could anyone help me or give me a hint please?
Long
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Closed Convex Sets of $L^1([0,1])$ and Minimal Norm

Let $M$ be the set of all $f\in L^1([0,1])$, relative to Lebesgue measure, such that $$\int_0^1f(t)\,dt=1.$$ Show that $M$ is a closed convex subset of $L^1([0,1])$ which contains infinitely many elements of minimal norm. For convexity - Let $f,g…
sjf2468
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Corollary of Hahn-Banach theorem

Let E be a normed linear space. Let $G \subset E$ a linear subspace. Show that if $g : G \to \mathbb{R}$ is a continuous linear operator, then $\exists f \in E^*$ such that $f_{/G} = g$ and $\|f\|_{E^*} = \|g\|_{G^*}$. Let $p : E \to \mathbb{R},…
g.pomegranate
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Minkowski functional $p_E$ is continuous if and only if $0\in E^0=\text{int}E$

Let $X$ be a normed topological vector space. Prove $p_E$ is continuous $\iff 0\in E^0$. In the above $p_E(x)=\inf\{t\ge0: x\in tE\},$ with $E$ an absorbing set $E\subset X$ is the Minkowski functional and $E^0$ denotes the interior of $E$. I…
alonso s
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Projection in Hilbert space onto non-closed subspace?

I know that if $H$ is a Hilbert space and $C$ a closed subspace one can define the orthogonal projection onto $C$ as the map $x \oplus y \in H = C \oplus C^\bot \mapsto x$. I am wondering: Is it valid, for an open subspace $O$, to define a…
user167889
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Question on extending densely defined linear operators on Hilbert space

Let $H$ be a Hilbert space. Is there a theorem that states that for a densely defined continuous linear operator $T: D(T) \subset H \to H$ there exists a unique continuous linear extension to $H$? And furthermore, If $T$ is an isometry then…
user167889
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Why is Isom(E,F) open in the set of bounded linear operators between E and F?

Let $ E $ and $ F $ be Banach spaces. According to the lecture notes I'm reading $ Isom(E,F) $ (the set of continuous isomorphisms between $ E $ and $ F $ with continuous inverse) is open in the set of bounded linear operators between $ E $ and $ F…
Ormi
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A Property about orthogonal projections

Let $H$ be a Hilbert space and $P_1,P_2$ orthogonal projections on the closed spaces $M_1$ and $M_2$. Let $\langle P_1x,x\rangle$ $\leq$ $\langle P_2x,x\rangle$ for all $x\in H$. Show that $P_1P_2=P_2P_1=P_1$ I am not able to get a connection…
Epsilondelta
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