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Given $X$=$c_0$, null sequence space with sup norm. Consider a subspace $Y$ of $c_0$ consisting of elements of $c_0$ as, $Y=\{x\in c_0 : x_{2i}=i.x_{2i-1}, i \geq 1\}$. I need to find the set of best …

It is known that in a reflexive Banach space, if the norm is strictly convex, then its dual will be smooth Banach space, and if the norm is smooth, then the dual norm is strictly convex. We can find a …

Consider the Hilbert space $l_2(\mathbb{N})$ under the square summable norm $\Vert \cdot \Vert_2.$ Let us define a new norm $||| \cdot ||| $ equivalent to $\Vert \cdot \Vert_2$ such that the closed un …

Let us consider the sequence space $c_0$ with the equivalent norm $$\Vert x \Vert^2 = \max_{i\ge1} \vert x^i \vert^2 + \sum_{i=2}^{\infty} 2^{-i+1} \vert x^i \vert^2 $$ for $x=(x^1,x^2,\ldots)\in c_0$ …

Consider the Hilbert space $l_2$ with an equivalent norm $$\Vert x \Vert = \max \{2 \Vert x \Vert_1, \Vert x \Vert_2 \},$$ where $\Vert x \Vert_1 =( \sum_{n=2}^\infty x_n^2 )^{\frac{1}{2}}$ and $\Vert …

Let $X=l_2$ with usual norm $\|\cdot\|_2$. We define a subspace of $X$ as $D=conv (B_{l_2} \cup B),$ where $B = \{ (x_n) \in l_2 : \sum_{n=1}^\infty \frac{n}{2} x_n^2 \leq 1\}$, conv is the convex com …

Let us consider the space $\ell_2$ with an equivalent norm defined by $$ \Vert x \Vert = \max \{ \Vert x^{'} \Vert_2, \Vert x^{''} \Vert_2 \}, $$ where $x^{'}=(0, x_2, x_3, \cdots)$, $x^{''} = (x_1, 0 …

Let us consider $\ell_2$ space with $\Vert \cdot \Vert_2$ norm. Let us define a new norm equivalent to $\Vert \cdot \Vert_2$ norm as follows: $$ \Vert x \Vert_0 = \max \{ \Vert x \Vert_2, \sqrt{2} \Ve …

First, we define the notion of strong sub-differentiability(SSD) of a norm on a Banach space $X$. The norm $\Vert \cdot \Vert$ of $X$ is said to be SSD if the one-sided limit $$\lim_{t \to 0+} \frac{\ …

Given $X$=$l^1$ and its dual space $X^*=l^\infty$. Now take $f=(1, 1/2, 2/3, 3/4,...) \in X^*$. Then clearly $\|f\|_\infty = 1$. I have found that $H=\ker f$ is a proximinal hyperplane in $X$. Note: A …

On the space $X=C[0,1]$, define a norm $||| f |||^2=\Vert f \Vert_{\infty}^2 + \Vert f \Vert_2^2$, where $\Vert \cdot \Vert_\infty$ is the sup norm on $C[0,1]$ space and $\Vert \cdot \Vert_2$ is the $ …

Let us take the Hilbert space $l_2$ with an equivalent norm $\Vert x \Vert = \max \{2 \Vert x \Vert_1, \Vert x \Vert_2 \}$, where $\Vert x \Vert_1 =( \sum_{n=2}^\infty x_n^2 )^{\frac{1}{2}}$ and $\Ver …

For an arbitrary set $\Gamma$, Day's norm on $c_0(\Gamma)$ is defined by $$ \Vert x \Vert = \sup \bigg \{ \bigg ( \sum_{k=1}^n 4^{-k} x^2(\gamma_k) \bigg )^{\frac{1}{2}} : (\gamma_1, \cdots, \gamma_n) …

We know that a Banach space $X$ is locally uniformly rotund(LUR) if for $x, x_n \in S_X$ with $\Vert x+x_n \Vert \to 2$, we have $x_n \to x$. In the same case, if $(x_n)$ has a convergent subsequence, …

Let us consider the space $\ell_2$ with the Hilbert norm $\Vert \cdot \Vert$ and consider the following eqivalent norm: $$ \Vert (r,x) \Vert_A^2 = \Vert (r, Tx)\Vert^2 + \max \{ \Vert x \Vert, \vert r …