Banach Spaces

Let \(V,W\) be Banach spaces, and let \(T_i:V\to W\) be a sequence of continuous linear operators such that \(\lim _{i\to \infty }T_i v\) exists for every \(v\in V\). Define the linear map \(T:V\to W\) by \[ Tv=\lim _{i\to \infty }T_i v \]

Show that \(T\) is continuous.

Prove that if \(A:\ell ^2(\mathbb {N})\to \ell ^2(\mathbb {N})\) is continuous and linear and \(AB(0,1)\) is dense in \(B(0,1)\) then \(A\) is surjective. Note that this is true for Banach spaces as well (though we are not asking you to prove it).

Let \((B,\|\cdot \|)\) be a Banach space. Explain the inclusion of \(B\) into its double dual. Prove that it is an isometry onto the image.

State the open mapping theorem for Banach spaces and use it to prove the following: Let \((B, \|\cdot \|_{B})\) and \((C, \|\cdot \|_{C})\) be Banach spaces and \(A : B \to C\) be continuous and linear. If \(A\) is a bijection then \(A^{-1}\) is continuous.

Let \(V\), \(W\) be Banach spaces, and let \(T_{i} : V \to W\) be a sequence of bounded linear operators such that \(\lim _{i \to \infty } T_{i}v\) exists for every \(v \in V\). Define the linear map \(T : V \to W\) by \[ Tv = \lim _{i \to \infty } T_{i}v. \]

Show that \(T\) is bounded.

Show that the Banach space \(\ell ^{p}\) with the usual norm is not a Hilbert space

when \(p \in [1, 2) \cup (2, \infty )\).

1. State the Closed Graph Theorem.
2. Let \(T : U \to V\) be a linear map between two Banach spaces such that, for any bounded linear functional \(f\) on \(V\) , the composite \(f \circ T\) is a bounded functional on \(U\) . Prove that \(T\) is bounded. You may use the fact that bounded linear functionals on a Banach space separate points.

1. State the Baire Category Theorem.
2. Let \(V\) be a Banach space, and \(T : V \to V\) a bounded linear map. Assume that for every \(v \in V\) there exists a non-negative integer \(n\) such that \(T^{n} (v) = 0\). Prove that there exists an integer \(n\) such that \(T^{n} (v) = 0\) for all \(v \in V\).

1. Give the definition of what it means for a sequence \((x_{n})\) in a Banach space \(X\) to converge weakly to \(x \in X\).
2. Let \(X = [0, 1]\) equipped with the Lebesgue measure. For \(n =\) 1, 2, …, let \(f_{n} = n \cdot \chi _{[0,1/n]}\), where \(\chi _{A}\) denotes the indicator function for the set \(A\). Prove that the sequence \(f_{n}\) does not converge weakly to 0 in \(L^{1}([0, 1])\).

Let \(C([0,1])\) be the Banach space of continuous functions on \([0,1]\) with the sup norm and view it as a natural subspace of \(L^\infty ([0,1])\), where \([0,1]\) is given Lebesgue measure, and the norm is essential supremum. Prove the existence of a bounded functional on \(L^\infty ([0,1])\) which is not identically \(0\) but vanishes on \(C([0,1])\).

Let \(X\) be a normed space and \(X^*\) be the space of bounded linear functionals. Every \(x\in X\) defines a functional \(\hat {x}\) on \(X^*\) by \(\hat {x}(f)=f(x)\) for all \(f\in X^*\). Prove that \(\lVert \hat {x}\rVert =\lVert x\rVert \).

Let \(V\) be a normed space and \(U\) a closed subspace. For every \(x\in V\), let \[ \lVert x+U\rVert =\inf \{\lVert x+y\rVert :\ y\in U\}. \] Then \(\lVert x+U\rVert \) is a norm on \(V/U\). (You do not need to prove that.) Assume \(U\ne V\). Prove that the map \(P:V\to V/U\) defined by \(P(x)=x+U\) has operator norm \(1\).

Let \(T:U\to V\) be a linear map between two Banach spaces such that, for any linear functional \(f\) on \(V\), the composite \(f\circ T\) is a continuous functional on \(U\). Prove that \(T\) is continuous. Hint: use the closed graph theorem.

Let \(V\) be a Banach space, and \(T:V\to V\) a bounded linear map. Assume that for every \(v\in V\) there exists a non-negative integer \(n\) such that \(T^n(v)=0\). Prove that there exists an integer \(n\) such that \(T^n(v)=0\) for all \(v\in V\).

Let \(X\) be a normed space, and let \(X^*\) be the space of bounded linear functionals on \(X\). For every \(x\in X\), let \(x^*\) be the functional on \(X^*\) defined by \(x^*(y)=y(x)\) for all \(y\in X^*\). Prove that \(x\mapsto x^*\) is an isometry from \(X\) into \(X^{**}\).

Let \(V\) be a normed space and \(U\) a closed subspace. Define the norm on \(V/U\) and prove that \(V/U\) is complete if \(V\) is.

Let \(V\) be a Banach space, and \(T:V\to V\) a bounded linear map. Assume that for every \(v\in V\) there exists a non-negative integer \(n\) such that \(T^n(v)=0\). Prove that there exists an integer \(n\) such that \(T^n(v)=0\) for all \(v\in V\).

Let \(T:U\to V\) be a linear map between two Banach spaces such that, for any linear functional \(f\) on \(V\), the composite \(f\circ T\) is a continuous functional on \(U\). Prove that \(T\) is continuous. Hint: use closed graph theorem.

Let \(V\) be a normed space and \(U\) a closed subspace. For every \(x\in V\), let \[ \lVert x+U\rVert =\inf \{\lVert x+y\rVert :\ y\in U\}. \]

(a) Prove that \(\lVert x+U\rVert \) is a norm on \(V/U\).

(b) If \(V\) is complete, so is \(V/U\).

Let \(T:V\to U\) be a bounded linear map between two normed spaces. Let \(T^*:U^*\to V^*\) be defined by \(T^*(f)=f\circ T\) for all \(f\in U^*\) (the adjoint map). Prove that \(\lVert T^*\rVert =\lVert T\rVert \).

Let \(V\) be a Banach space, and \(T:V\to V\) a bounded linear map. Assume that for every \(v\in V\) there exists a non-negative integer \(n\) such that \(T^n(v)=0\). Prove that there exists an integer \(n\) such that \(T^n(v)=0\) for all \(v\in V\).

Let \(T:V\to W\) be a bounded linear map, where \(V\) and \(W\) are normed spaces. Let \(U\) be the kernel of \(T\). Prove that \(\lVert S\rVert =\lVert T\rVert \) where \(S:V/U\to W\) is such that \[ S(x+U)=T(x) \] for all \(x\in V\).

Let \(V\) be a Banach space, and \(T:V\to V\) a bounded linear map. Assume that for every \(v\in V\) there exists a non-negative integer \(n\) such that \(T^n(v)=0\). Prove that there exists an integer \(n\) such that \(T^n(v)=0\) for all \(v\in V\).

Prove or give a counterexample: \(B_1,B_2\) are Banach spaces and \(A:B_1\to B_2\) is continuous, linear, injective, and \(A(B_1)\) is dense. Then \(A\) is a bijection.

Prove the Banach-Steinhaus theorem. That is, show that if \((B,\lVert \cdot \rVert _B)\) and \((\hat B,\lVert \cdot \rVert _{\hat B})\) are Banach spaces and \(A_i:B\to \hat B\) are a sequence of continuous linear maps from \(B\) to \(\hat B\), then either there exists \(M\) so that \(\lVert A_i\rVert _{\mathrm {op}}<M\) for all \(i\), or there exists \(x\in B\) so that \[ \sup _i\lVert A_i x\rVert _{\hat B}=\infty . \]

Prove the Baire category theorem. That is, show that if \((X,d)\) is a complete metric space and \(U_1,U_2,\ldots \subset X\) are open and dense, then \(\cap _{i=1}^{\infty }U_i\ne \emptyset \).

Let \(V\) and \(W\) be Banach spaces, and \(L(V,W)\) the space of bounded linear maps from \(V\) to \(W\). Suppose that \(T_n\in L(V,W)\), \(n=1,2,\ldots \), and that \(\lim T_nx\) exists for each \(x\in V\). Show that \(T:V\to W\) defined by \(Tx=\lim T_nx\) is in \(L(V,W)\).

(a) Let \(\ell ^1\) and \(\ell ^\infty \) be the Banach spaces of sequences of complex numbers \(\xi =(\xi _1,\xi _2,\ldots )\) such that \[ \xi \in \ell ^1 \quad \text {if and only if}\quad \lVert \xi \rVert _1:=\sum _{i=1}^{\infty }|\xi _i|<\infty \] and \[ \xi \in \ell ^\infty \quad \text {if and only if}\quad \lVert \xi \rVert _\infty :=\sup _i|\xi _i|<\infty . \] Let \((\ell ^\infty )^*\) denote the Banach space of continuous linear functionals, equipped with the operator norm, on \(\ell ^\infty \). Show that the map \(\Lambda :\ell ^1\to (\ell ^\infty )^*\) defined by \[ \Lambda (\xi )(\eta _1,\eta _2,\ldots )=\sum _i \xi _i\eta _i \] is a well-defined norm-preserving injection of \(\ell ^1\) into \((\ell ^\infty )^*\) which is not onto.

Hint: Show that \((\ell ^\infty )^*\) contains nontrivial functionals which vanish on the subspace consisting of those \(\xi \) such that \(\xi _i\to 0\) as \(i\to \infty \).

(b) Let \((X,\mu )\) be a measure space. State precise conditions on \(\mu \), \(p\), and \(q\) which guarantee a norm-preserving isomorphism of \(L^p(X)\) onto \((L^q(X))^*\).

Let \(X\) and \(Y\) be Banach spaces, and suppose that \(T:X\to Y\) is a compact surjective linear map. Show that \(Y\) is finite dimensional. (Recall that a linear map is compact if the image of any bounded set has compact closure.)

Let \(\ell ^1\) be the Banach space of real sequences \(a=(a_1,a_2,\ldots )\) with norm \[ \lVert a\rVert _1=\sum _{i=1}^{\infty }|a_i|<\infty , \] and \(\ell ^\infty \) the Banach space of real sequences with norm \(\lVert x\rVert _\infty =\sup _i |x_i|\). Let \(a\in \ell ^1\) be a fixed sequence; for any sequence \(x\in \ell ^\infty \), define a new sequence \(T_a(x)\) by \[ \bigl (T_a(x)\bigr )_n=\sum _{i=1}^{n}a_i x_i \]

(a) Show that this defines a bounded operator \(T_a:\ell ^\infty \to \ell ^\infty \).

(b) Show \(\lVert T_a\rVert =\lVert a\rVert _1\), where the left-hand side denotes the norm of the operator \(T_a:\ell ^\infty \to \ell ^\infty \).

Let \(\ell ^\infty \) denote the Banach space of bounded real sequences with norm \(\lVert x\rVert _\infty =\sup _i |x_i|\), and \(\ell ^1\) the Banach space of real sequences with norm \(\lVert x\rVert _1=\sum _i |x_i|<\infty \). For \(x\in \ell ^\infty \) and \(y\in \ell ^1\) define \(Tx(y)=\sum _i x_i y_i\). Prove that \(T:\ell ^\infty \to (\ell ^1)^*\) is a bounded operator that is one to one and onto. (Here \((\ell ^1)^*\) is the dual space of \(\ell ^1\).)

Prove that if \(X\) is a Banach space with a closed unit ball which is compact, then \(X\) must be finite dimensional. Hint: consider the family of sets of the form \(\{x\in X:\|x\|=1,\text { and }f(x)=0\}\) where \(f\) is a bounded nonzero linear functional on \(X\).

Let \(\alpha (x,y)\) be a continuous function on \([0,1]\times [0,1]\). Show that \[ A(f)(x)=\int _0^1 \alpha (x,y)f(y)\,dy \] defines a bounded operator from \(L^2([0,1])\) to \(C([0,1])\), considered as a Banach space with respect to \(\|f\|=\sup _x |f(x)|\).

Let \(X\) be a Banach space and let \(T\) be a function which assigns to each real number \(s\in \mathbb R\) a bounded linear operator \(T(s):X\to X\). The derivative \(T'(s)\) of \(T\) at \(s\) is the operator defined by \[ T'(s)x=\lim _{h\to 0}h^{-1}(T(s+h)-T(s))x \] where the domain of \(T'(s)\) is the set of all \(x\in X\) for which this limit exists. Prove that if the domain of \(T'(s)\) is all of \(X\) for a given \(s\), then \(T'(s)\) is a bounded linear operator on \(X\).

Let \(X\) be a compact Hausdorff space and denote by \(C(X)\) the Banach space of continuous complex valued functions on \(X\) and by \(P(X)\) the cone consisting of all \(f\in C(X)\) such that \(\operatorname {Re}(f)\ge 0\). If \(H\) is a linear subspace of \(C(X)\) such that \(H\cap P(X)=(0)\), prove that

(a) \(\|\lambda +h\|_\infty \ge |\lambda |\) for all \(\lambda \in \mathbb C\), \(h\in H\) (hint: show this is equivalent to \(\|1-h\|_\infty \ge 1\) for all \(h\in H\));

(b) there exists a positive Borel measure \(\mu \) on \(X\) such that \(\mu (X)=1\) and \[ \int h\,d\mu =0 \quad \text {for all } h\in H \] (recall that a measure \(\mu \) on \(X\) is positive if \(|\mu |=\mu (X)\)).

Let \(X\) be a compact Hausdorff space. Let \(C(X)\) be the space of complex valued continuous functions on \(X\). We assume that \(C(X)\) is equipped with a norm \(\|\cdot \|\), which turns it into a Banach space. Assume moreover, that, for each \(x\in X\) the linear functional \(\lambda _x\), defined by: \(\lambda _x(f)=f(x)\) is bounded on this Banach space. Show that there exist positive constants \(A\) and \(B\), such that for every \(f\in C(X)\) \[ A\|f\|_\infty \le \|f\|\le B\|f\|_\infty . \] Here \(\|f\|_\infty \) denotes, as usual, the sup norm.

A bounded linear transformation \(A\) from a Banach space \(X\) to a Banach space \(Y\) is said to be ”compact” if the image under \(A\) of the unit ball in \(X\) has compact closure in \(Y\). Prove that if a compact linear transformation is also surjective, then \(Y\) is finite dimensional (you may assume that a locally compact Banach space must be finite dimensional).

Let \(X_1,X_2\) be Banach spaces, \(T\) a linear map of \(X_1\) into \(X_2\), and \(\{\lambda _\alpha \}_{\alpha \in A}\subset X_2^*\) be a subset which separates points of \(X_2\); that is, if \(v_1,v_2\in X_2\), \(v_1\ne v_2\), then there exists \(\alpha \in A\) such that \(\lambda _\alpha (v_1)\ne \lambda _\alpha (v_2)\). Suppose that \(\lambda _\alpha \circ T\in X_1^*\) for all \(\alpha \in A\). Then show that \(T\) is continuous.

Let \(\ell ^\infty \) be the normed space consisting of all complex valued, bounded sequences \[ \mathbf a=(a_1,a_2,\cdots a_n\cdots ) \] with the norm \(\lVert \mathbf a\rVert =\sup \{|a_n|:n\in \mathbb N\}\). Let \(W\subset \ell ^\infty \) be the subspace consisting of all \(\mathbf a\in \ell ^\infty \) such that \(\{n\in \mathbb N:a_n\ne 0\}\) is a finite set.

Show that there exists a nontrivial bounded linear functional on \(\ell ^\infty \) which vanishes on \(W\).

Let \(B\) be a Banach space and \(A\subset B\) a closed subspace. If \(x\in B\) but \(x\notin A\), prove that there is a bounded linear functional \(f\) on \(B\) such that \(f(x)=1\), \(f(y)=0\) for every \(y\in A\) and \(\lVert f\rVert =(\inf \{\lVert x+y\rVert :y\in A\})^{-1}\).

Let \(A\), \(B\), and \(C\) be Banach spaces and let \(f:A\to C\) and \(g:B\to C\) be bounded linear maps. Suppose that \(g\) is one to one and \(f(A)\subset g(B)\). Prove that there is a bounded linear map \(h:A\to B\) such that \(b=h(a)\) if and only if \(f(a)=g(b)\).