Mathematics Stack Exchange
Mathematics Stack Exchange

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
2
votes
1 answer

A question concerning quotient of Banach spaces

Let $X$ be a Banach space and $W$ be a proper closed subspace. Let $0 < \epsilon < 1$. Since $X/W$ is nonzero, one can find some $v \in X$ such that $\|v + W\| = \epsilon$. But then it is possible to find some $w \in W$ such that $\|v - w\| = 1$?…
Ken Leung
  • 335
2
votes
1 answer

Sequence spaces with the standard Schauder basis

The standard Schauder basis for $l^p$ spaces with $1\leq p < \infty$ is $e_n = \{\delta_{n,j}\}_{j=1}^\infty$ so $$ e_1=(1,0,0,...)\\ e_2=(0,1,0,...)\\ ... $$ $l^\infty$ doesn't have a Schauder basis because it's not separable. Are there any…
Konstantin
  • 2,003
2
votes
1 answer

Convergence in generalized functions(Distributions)

I know that convergence in $D'(\Omega)$ (The space of distributions) is just the convergence in $\mathbb{R}$ for every test function $\phi$. But I am not sure about something like " Norm convergence implies convergence is distributions" , what about…
Mathslover shah
  • 537
2
votes
1 answer

Exercises 13 of Chapter 15 in Lax's Functional analysis (Sets of second category)

Could anyone give some idea about the proof of the following theorem?
Xianjin Yang
  • 677
2
votes
1 answer

The norm of operators in Hilbert space

In this question $\mathcal{H}$ stands for a Hilbert space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, with inner product $\langle\cdot\;| \;\cdot\rangle$ and the norm $\|\cdot\|$ and let $\mathcal{B}(\mathcal{H})$ the algebra of all bounded linear…
Student
  • 4,914
2
votes
1 answer

Prove $f(x,y)$ is a contraction in $(\mathbb(R^2),||.||_{1})$

We define the norm in $\mathbb{R^2}:$ by: $$\forall (x,y)\in\mathbb{R^2}, ||(x,y)||=|x|+|y|$$ Let $f$ be defined in $\mathbb{R^2}$ and $$f(x,y)=\left(\frac{1}{4}\sin(x+y),1+\frac{2}{3}\arctan(x-y)\right)$$ Prove $f(x,y)$ is a contraction in…
liverpool29
  • 197
2
votes
1 answer

Show that there is $m>0$ such that $\|T(x)\|\geq m\|x\|$ for all $x\in X$ iff $T$ is injective and $T(X)$ is closed.

Let $X,Y$ be Banach spaces and $T:X\to Y$ be a bounded linear operator. It is required to show that there is a constant $m>0$ such that $\|T(x)\|\geq m\|x\|$ for all $x\in X$ if and only if $T$ is injective and $T(X)$ is closed. I proved the…
Janitha357
  • 3,049
2
votes
2 answers

proof of a Banach space

Let X be a Banach space with the norm $||.||$ and f be a vector function which acts from $\mathbb {R}$ into the Banach space X. Can you please show that the following space is a Banach space…
tuma
  • 21
2
votes
1 answer

Dual cone of $\mathbb{R}^{n}$

Let $E$ be a Banach space and $P\subset E$ be a cone. A nonempty subset P of a Banach space E is called a cone if P is convex, closed, and satisfies the conditions: (i) $λx\in P$ for all $x\in P$ and all real positive number $λ$. (ii) $x,−x\in P$…
sbl
  • 51
2
votes
2 answers

Why does $|T^{*}T|\geq |T|^2$ hold for the operator norm?

I'd like to know how to prove that $|T^{*}T|=|T|^2$. I know that is $\leq$ since $|T|=|T^{*}|$ but I don't know how to prove the reverse inequality. Thanks.
Algge
  • 23
2
votes
3 answers

Topologically equivalent metrics

Can a topological vector space have two metrics both of which give rise to same topology as the space has, but one is translation invariant but the other is not?
Nirakar Neo
  • 889
2
votes
1 answer

Quotient space and continuous linear operator.

Let $X$ and $Y$ be banach space let, $T\in B(X,Y)$. Define $N$ as the Kernel of $T$ and $M$ as the range of $T$ . We know that $\bar T : X/N\to M $ is well defined , ie if $(x+N) =(y+N)$ then $Tx$ and $Ty$ are equal , and this map is one to one ,…
Theorem
  • 7,979
2
votes
0 answers

Is $\mathcal{C}^\infty([0,1])$ normable?

The space $\mathcal{C}^\infty(]0,1[)$ of smooth real functions over $]0,1[$ is not normable (because it is infinite dimensional and Montel). Consider instead the space $\mathcal{C}^\infty([0,1])$ of smooth real functions over $[0,1]$. Here, wiki…
Noé AC
  • 1,598
  • 10
  • 14
2
votes
1 answer

Separability, Boundedness and Isomorphism in Banach space.

I am trying to solve the following problem but I don't really getting it where to start from, which way to think. Any help would be appreciated. If $(X,\|.\|)$ is a separable Banach space, a) Why is the unit sphere $S=\{x\in X ;\|x\|=1\}$…
Theorem
  • 7,979
2
votes
1 answer

Why does using the Closed Graph Theorem make it easier to show boundeness of a linear operator?

I read that showing $A$ is bounded by showing its graph is closed is 'more efficient' or 'easier' than showing it is bounded directly. For a specific example, consider the Hellinger-Toeplitz Theorem states that if $H$ is a Hilbert space and $A:H\to…
ManUtdBloke
  • 2,594
1 2 3
…
99 100