---
title: "Some questions about a special semiscalar product"
description: "Define the semiscalar product [x,y] by $ $ [x,y]=\inf_{t>0}\frac{1}{2t}[||x+ty||^2-||x||^2].$ $ E be an n.v.s. I donot know how to prove that 1.$ [x,\lambda x+\mu y]=\lambda||x||^2+\mu[x,y]\..."
url: https://extraproxies.com/some-questions-about-a-special-semiscalar-product
date: 2023-10-18
modified: 2023-10-18
author: "ExtraProxies"
categories: ["Proxy Feed"]
tags: ["about", "product", "questions", "semiscalar", "Some", "special"]
type: post
lang: en
---

# Some questions about a special semiscalar product

Define the semiscalar product by $ $ =\inf_{t>0}\frac{1}{2t}[||x+ty||^2-||x||^2].$ $ E be an n.v.s. I donot know how to prove that

1.$ =\lambda||x||^2+\mu\ \forall x,y\in E,\lambda\in\mathbb{R},\forall \mu >0.$

2.$ [\lambda x,\mu y]=\lambda\mu,\forall x,y\in E,\forall\lambda,\mu\geq 0$

I probably understand that this semiscalar product is similar to an inner product, but I only have trigonometric inequalities for its norm, how do I prove that this product is linear?