On 10th June 1940 , Italy declarted war on France and Britain. On 26 May, Mussolini informed Italian Marshals Pietro Badoglio, chief of the Supreme General Staff, and Italo Balbo that he intended to join the German war against Britain and France, so to be able to sit at the peace table “when the world is to be apportioned” following an Axis victory. The two marshals unsuccessfully attempted to persuade Mussolini that this was not a wise course of action, arguing that the Italian military was unprepared, divisions were not up to strength, troops lacked equipment, the empire was equally unprepared, and the merchant fleet was scattered across the globe. On 5 June, Mussolini told Badoglio, “I only need a few thousand dead so that I can sit at the peace conference as a man who has fought”.
The two sides exchanged air raids on the first day of war, but little transpired on the Alpine front, since France and Italy had defensive strategies. There was some skirmishing between patrols and the French forts of the Ligne Alpine exchanged fire with their Italian counterparts of the Vallo Alpino. On 17 June, France announced that it would seek an armistice with Germany. On 21 June, with a Franco-German armistice about to be signed, the Italians launched a general offensive along the Alpine front, the main attack coming in the northern sector and a secondary advance along the coast. The Italians penetrated a few miles into French territory against strong resistance but stalled before its primary objectives could be attained, the coastal town of Menton being the most significant conquest.
On the evening of 24 June, an [armistice was signed at Rome. It came into effect just after midnight on 25 June, at the same time as the armistice with Germany signed 22 June. Italy was allowed to occupy the territory it had captured in the brief fighting, a demilitarised zone was created on the French side of the border.
On Mediterranean Sea , action started between Royal Navy and Italian Navy and Air Force that would involve Germans later. On June 1940 :
from https://www.naval-history.net/WW2CampaignsRNMed.htm
Malta - Italian aircraft carried out the first of the many raids on Malta on the 11th. Next day, the RAF made its first attacks on Italian mainland targets.
12th -Royal Navy Mediterranean Fleet with “Warspite”, “Malaya”, “Eagle”, cruisers and destroyers sailed from Alexandria for a sweep against Italian shipping in the Eastern Mediterranean. South of Crete, light cruiser “CALYPSO” was torpedoed and sunk by Italian submarine “Bagnolini”.
13th - Mediterranean Fleet submarines operated out of Alexandria on patrol off Italian bases and soon lost three of their number . At the time mines were usually blamed, but it turned out Italian anti-submarine forces were far more effective than expected. The first loss was “ODIN” off the Italian coast in the Gulf of Taranto, sunk by the guns and torpedoes of destroyer “Strale”.
16th - The second British submarine “GRAMPUS” ,minelaying off Augusta, Sicily was caught and sunk by large torpedo boats “Circe” and “Clio”.
17th - Six Italian submarines were sunk in the Mediterranean, half by the Royal Navy. However the first to go, “PROVANA” was rammed and sunk off Oran, Algeria by French sloop “La Curieuse” after attacking a French convoy, and just a week before France was forced out of the war.
19th - Towards the other end of the North African coast, the third British loss “ORPHEUS” was sent to the bottom by Italian destroyer “Turbine” north of the Cyrenaica port of Tobruk, soon to become a household name .
20th - The second Italian boat lost in the Mediterranean was “DIAMANTE” torpedoed by submarine “Parthian” off Tobruk.
27th - The second Italian submarine lost was the “LIUZZI” sunk by Royal Navy Med Fleet destroyers “Dainty”, “Ilex”, “Decoy” and the Australian “Voyager” south of Crete.
28th - As the Mediterranean Fleet 7th Cruiser Squadron covered convoy movements in the Eastern Mediterranean, three Italian destroyers carrying supplies between Taranto in southern Italy and Tobruk were intercepted. In a running gun battle, Italian destroyer “ESPERO” was sunk by Australian cruiser “Sydney” to the southwest of Cape Matapan at the southern tip of Greece.
28th - The first of two Italian submarines sunk by RAF Sunderlands of No. 230 Sqdn was “ARGONAUTA” in the central Med as she was believed to be returning from patrol off Tobruk
29th - The same Med Fleet destroyers after sinking “Liuzzi” two days earlier, were now southwest of Crete. They repeated their success by sinking “UEBI SCEBELI” .
29th - A day after their first success, the Sunderlands of No. 230 Sqdn sank “RUBINO” in the Ionian Sea as she returned from the Alexandria area
British Force H - By the end of the month, Force H had been assembled at Gibraltar from units of the Home Fleet. Vice-Adm Sir James Somerville flew his flag in battlecruiser “Hood” and commanded battleships “Resolution” and “Valiant”, carrier “Ark Royal” and a few cruisers and destroyers. He reported directly to the Admiralty and not to the Commander, North Atlantic. From Gibraltar, Force H could cover the Western Mediterranean and the Atlantic, as happened in the May 1941 hunt for the “Bismarck”. Units could also quickly transfer back to the Home Fleet and UK waters as shortly became necessary at the height of the German invasion scare. There could be no better example of the flexibility of British naval power at this time.
Warship Loss Summary - In a confusing month, the Royal Navy had lost one light cruiser, one destroyer, three submarines and one sloop; the Italian Navy one destroyer and ten submarines, including four in the Red Sea.
Merchant Shipping War - Losses in the Mediterranean throughout the war would generally be low as most Allied shipping to and from the Middle East was diverted around the Cape of Good Hope.
Monthly Loss Summary
6 British, Allied and neutral ships of 45,000 tons from all causes.
On Libya-Egypt frontier , on 11th and 12nd June 1940 , British Royal Armored Car squadrons from 7th Motorised Brigade raided across border into Cyreneica deep into Italian territory , attacked and captured Italian fortresses at Capusso and Sollum along with 300 Italian prisoners. Most Italian POWs were unaware that war was declared between Britain and Italy and couldn’t hide their suprise.