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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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Prove that $\lim_{n\rightarrow \infty} T_n(x) = T(x) \iff \lim_{n\rightarrow \infty}\sup_{x \in K}\|T_n(x) - T(x)\| = 0$

Suppose $X$ and $Y$ are Banach spaces and $T:X \rightarrow Y$ is a BLO and $K$ is a compact subset of $X$. Prove that: $$\lim_{n\rightarrow \infty} T_n(x) = T(x) \iff \lim_{n\rightarrow \infty} \sup_{x \in K}\|T_n(x) - T(x)\| = 0$$ the…
Kees Til
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proving an element is unitary in a C* algebra

Let $p,q$ be projections in a unital C*-algebra $A$ and let $\tilde{A}$ be the unitization. I'd like to show that if $p\sim_u q$ (ie $q=zpz^*$ for $z$ unitary in $\tilde A$), then $q=upu^*$ for $u$ unitary in $A$. The book proceeds as follows: Set…
Jake
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Let $F$ a separable space and $T: E \to F$ a linear isometry. Is $E$ a separable space?

Let $F$ a separable space and $T: E \to F$ a linear isometry. Is $E$ a separable space? (E, F are linear spaces) I was working in the following problem: Showing that $T: l_\infty \to L(l_2,l_2)$ defined by $T(a) (b) = (a_n b_n), \quad a = (a_n) \in…
user 242964
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is this an counterexample for: $(C[a,b],\| \cdot \|_2)$ is complete?

our prof wanted to show that $(C[0,1],\| \cdot \|_2)$ is not complete. So he said $$f_k(x) = x^k$$ is a counterexample. I wonder if this is true. I tried to show that $f_k$ is cauchy sequence. But i ask myself if the limit is $f \equiv 0$ or $f =…
user317721
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Can a topological vetor space over $\mathbb{R}$ or $\mathbb{C}$ ever be considered "first category"?

I'm given the exercise to show that a finite dimensional linear subspace of an infinite topological vector space $X$ is nowhere dense (which I can do), and then to show that if $X$ is the union of countably many finite dimensional linear subspaces,…
user140776
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Problem with a proof on Conway's book

Now assume that $C_b(X)$ is separable. Thus $({\rm ball}\,C_b(X)^*,{\rm wk}^*)$ is metrizable (5.1). Since $X$ is homeomorphic to a subset of ${\rm ball}\,C_b(X)^*$ (6.1), $X$ is metrizable. It also follows that $\beta X$ is metrizable. It must be…
MickG
  • 8,645
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Dual space is isometrically isomorphic

Let $\{X_i:i\in I\}$ be a collection of normed spaces. If $1\leq p < \infty$, show that the dual space of $\bigoplus_p X_i$ is isometrically isomorphic to $\bigoplus_q {X_i}^\ast$, where $1/p+1/q=1$. I know that if X is normed space, then $X^\ast$…
asfajaf
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Show the operator $T$ is bounded if and only if $\sup|\lambda_j| < \infty$

Let $(\lambda_n)$ be a sequence of non-zero scalers and let $D(T)= \{x=(\epsilon_j) \in l^2 : \sum^\infty_{j=1} |\lambda _j |^2 |\epsilon _j |^2 <\infty \}$ We define a linear operator $D(T) \to Ran(T)$, $D(T) \in l^2$ and $Ran(T) \in l^2$ ,…
Al jabra
  • 2,331
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Functional and Linear Functional

may I know what is the distinction between functional analysis and linear functional analysis? I do a search online and came to a conclusion that linear functional analysis is not functional analysis and am getting confused by them. When I look at…
Sandra
  • 563
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Can $C^\infty(\mathbb{T})$ become a Banach space?

Let $T$ be the unit circle and $C^\infty(\mathbb{T})$ the set of functions defined on $\mathbb{T}$ which have derivatives of every order. I know that $C^\infty(\mathbb{T})$ with the metric induced by the seminorms…
Chris Apostol
  • 555
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Proving that $f = 0$ if $\int fg = 0$ for all $g \in S$

Let $f \in L^1$. I want to prove that if $\int f g = 0$ for all $g\in S$, then $f = 0$ a.e. $S$ denotes Schwartz space. My Approach: My idea is to let $h = sgn(f)$ and then smooth it somehow to get a function in $S$. But I don't know how to proceed.…
Mehdi Jafarnia Jahromi
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I don't understand Conway's Banach limit proof

In Conway's functional analysis text, he claims that there is a linear functional $L:l^{\infty}\rightarrow \mathbb{R}$ such that if $x\in c$, $L(x)=\lim x_{n}$. Here $c$ will denote the space of sequences that converge to a point and $c_{0}$ the…
JessicaK
  • 7,655
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sum of open balls in normed space

Let $X$ be a normed nonempty space and $x \in X$. We define the set:$$B(x,r)=\{y \in X:\|y-x\|
user308560
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checking whether an operator is bounded or not

let X be a real normed space with finitely many non zero terms,with supremum norm and let T:X$\to$X be a one-one and onto linear operator defined by $$T(x_1,x_2,x_3,......)=(x_1,\frac{x_2}{4},\frac{x_3}{9},.....)$$ then, which of the following is…
Abhishek Shrivastava
  • 854
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Infinite intersection of kernels

Let $X$ be an infinite dimensional Banach space. And let $X^{\mathrm{*}}$ be the space of linear continuous functionals( $f : X \rightarrow \mathbb{R}$ linear and continuous). Assume $X^{\mathrm{*}}$ is separable, and take $(x^{\mathrm{*}}_n)$for $n…
clark
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