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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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Equivalence of the Two Definitions of Norm of Dual Space

The dual space of a normed linear space $V$ is the space of all linear bounded functional on $V$: $$ V^*:=\{f:V\to R\mid\text{$f$ is linear and bounded}\} $$ The norm of $V^*$ is defined as: $$ \|f\|=\sup_{\|u\| \leq 1} |f(u)| \tag{1} $$ Can you…
A Slow Learner
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Can we define 'Norm' over an arbitrary vector space which may not be real or complex vector space?

While we define norm on a vector space, we consider only real or complex vector field. But can we generalize this norm on a vector space over an arbitrary field ? I think this can be done, but we have to define a suitable modulus function on the…
SJA
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Examples of some linear and nonlinear operators

Let $H$ be a Hilbert space. Could you please give me examples of linear or nonlinear operators $F: H \to H$ such that $$ \limsup\limits_{\|x-y\| \to 0} \|F(x)-F(y)\| = +\infty \quad \forall x,y\in H ? $$
Boris Petrov
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Why does setting y=x work in this proof?

The problem in question asks you to show that the non-negativity of a metric follows from the 2nd, 3rd, and 4th metric space axioms i.e. $\textbf{M2}$ $d(x,y) = 0$ iff $x=y$ $\textbf{M3}$ $d(x,y) = d(y,x)$ $\textbf{M4}$ $d(x,y) \leq d(x,z) +…
Henry Powell
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Trace norm of a trace class operator exercise in Conway

Exercise in Conway's Functional Analysis book: Let $T$ be a trace class operator on a Hilbert space ${\cal H}$. Prove: $$\sup\{|\mbox{tr}(CT)|:\ C\ \mbox{is compact}, ||C||\leq 1\}=||T||_1.$$ Here, $||T||_1=\mbox{tr}[(T^*T)^{\frac{1}{2}}]$ is the…
chhro
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Is the closed unit ball of a Hilbert space a weak sequential space?

I am currently reading a proof on characterizations of a compact operator for Hilbert spaces. Let $\mathscr{B}_1$ be the closed unit ball of the Hilbert space $H$. One of the equivalent statement involves: $T\in B(H)$ is the norm-limit of finite…
Kurome
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Let $f$ be a bounded linear operator $X \to X$ s.t that $\|f(x)\|\geq m\|x\|$ for some $m$, $\forall x \in V$. Prove that $f$ cannot be compact.

Let $V$ is an infinite dimensional subpace of a Banach space and let $f$ be a bounded linear operator $X \to X$ s.t that $\|f(x)\|\geq m\|x\|$ for some $m$, $\forall x \in V$. Prove that $f$ cannot be compact. My solution relies ont he fact that we…
2132123
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If $T:X\to X$, $X$ is a Banach space, and $T$ linear, maps closed sets onto closed sets. Can I say that $T$ has a closed graph?

I know if $X$ were to incomplete space such as $c_{00}$ $X$ is a normed vector space and $T:X\to X$ is a function that has a closed graph, does $T$ map closed sets to closed sets? the hypothesis does not hold but what about if $X$ were to be a…
Jale'de jaled
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Find a linear operator $T:X\to X$ , $X$ normed space and $T$ maps closed sets onto closed sets, but $T$ is not bounded.

I sense the completness of the normed spaces has a role, so I searched the incomplete metric spaces such as $c_{00}$ to find examples. And space must be infinite dimensional otherwise since every finite dimensional normed space is Banach I may not…
Jale'de jaled
  • 780
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Finding norm of $Tf(t)=\int_0^1k(s,t)f(s)ds$ if $k$ is continuous functional defined on a unit square and $T:C[0,1]\to C[0,1]$

Since $k$ is continuous and defined on $[0,1]\times[0,1]=J$ compact domain so it is bounded. Therefore there exists $c\in \mathbb R$ such that $|k(s,t)|\le c,\quad \forall (s,t)\in [0,1]\times[0,1]=J$ So $$||Tx||=\sup_{t\in…
Jale'de jaled
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Integral operator on $L^p$

The problem is : (Conway's exercise in 'The Open Mapping and Closed Graph Theorems' section) (Transcribed from this image) Let $(X, \Omega, \mu)$ be a $\sigma$ -finite measure space, $1 \leqslant p<\infty,$ and suppose that $k: X \times X…
Silement
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Parseval identity to compute $ \int_{-\infty}^{\infty} (\frac{\sin x}{x})^2dx$

The Parseval identity in Hilbert space $H$ is If $h \in H$, $\lVert h\rVert = \sum \{||^2:e \in S\} $ then when $S$ is a basis for $H$ How to use this to compute $$ \int_{-\infty}^{\infty} \left(\frac{\sin x}{x}\right)^2dx$$
Silement
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Does $C_{c}(\mathbb{R})$ have a countable Hamel basis?

I was wondering if $C_{c}(\mathbb{R})=\text { continuous functions with compact support }$ have countable Hamel basis. My intuition tells me try to use Baire Category Theorem and I found one corollary saying that for an infinite-dimensional Banach…
Maskoff
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Help following an outline of Markov–Kakutani Fixed Point Theorem proof.

I'm trying to prove Markov–Kakutani Fixed Point Theorem by following the outline on page 173 here: https://people.math.ethz.ch/~salamon/PREPRINTS/funcana-ams.pdf Let X be a locally convex Hausdorff topological vector space and let $A$ be a…
Muselive
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Show that a subset of a normed space is bounded iff its image by any operator is bounded

Could you give me a hint with the following exercise? Let $X$ be a $\mathbb{K}$-normed vector space and $Y\subset X$, show that $Y$ is bounded respect to the norm on $X$, iff, for all $T\in X'$, $T(Y)$ is bounded on…
Framate
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